
\phantomsection\label{contents}
\pdfbookmark[1]{Contents}{contents}
\setcounter{tocdepth}{4}
\tableofcontents



\section{1~~~Why do we need SUSY ?%
  \label{why-do-we-need-susy}%
}


\subsection{1.1~~~Drawbacks and unsolved problems of Standard Model%
  \label{drawbacks-and-unsolved-problems-of-standard-model}%
}

Despite of Standard Model (SM) tremendous success, the theory
is not believed to be theoretically satisfactory, and is
regarded as low-energy effective theory of a more fundamental
theory. There are many reasons for the conceptual incompleteness:
the assignment of the quantum numbers to the fermions
is not fully clear; why  whitin SM      the bulk matter is neutral;
we don't understand why there are three apparantly unrelated
gauge groups and what rules the strength of their
coupling constants; there is no a sufficient reason for
which there are only  three generations of fermions; it's not explained
why the fermion mass spectrum ranges over thirteen orders of magnitude.
Moreover, the presence of the scalar field in SM is completely
artifical, since it's introduced just for the purpose of breaking
the electroweak symmetry. There is only one scalar boson while all other bosons are vectors in the theory.
The problem of CP-violation described by \DUrole{raw-tex}{$\theta_{QCD}$} phase   is not well understood including CP-violation in a strong
interaction. Another question is related to flavour mixing and the number of generations which are arbitrary.
In the strict framework of the SM, nuetrinos are massless.
However, there is an experimental evidence that neutrinos
must be massive particles, although very light ones.

The Standard Model depends on nineteen parameters: the three gauge coupling constants, the two parameters \DUrole{raw-tex}{$\mu^{2}$} and
\DUrole{raw-tex}{$\lambda$} which determine the mass and the self coupling of the
Higgs field, the nine quark and charged lepton masses, the three
angles and one phase specifying the quark mixing matrix, the
\DUrole{raw-tex}{$\theta_{QCD}$} phase related to strong spontaneous  CP violation \DUrole{raw-tex}{\cite{Dine:2000cj},\cite{Kim:2008hd}}
\DUrole{raw-tex}{$$\frac{\theta_{QCD}}{32\pi^2}g_S^2\tilde{G}^{a,\mu\nu}G_{\mu\nu}^a,$$}
where \DUrole{raw-tex}{$ G^{a}_{\mu\nu}$} is the strength tensor of gluon field and
\DUrole{raw-tex}{$\tilde{G}^{a}_{\mu\nu}$} is the axial strength tensor \DUrole{raw-tex}{$$\tilde{G}^{a}_{\mu\nu}=\epsilon_{\mu\nu\alpha\beta}G^{a,\alpha\beta}. $$}
Moreover, many more parameters are needed to accommodate non-accelerator observations \DUrole{raw-tex}{\cite{Ashie:2004m}, \cite{Ahmad:2002jz}}.
The results from the Super Kamiokande experiment on the leptons observed in the atmospheric showers of particles stimulated by cosmic rays incident on the top of the atmosphere seem
to clearly indicate that the muon neutrino exhibits oscillatory behavior. This can interpreter neutrinos as Majorana particles
requiring  three neutrino masses, three mixing angles and one CP-violating phase.
Most physicists believe that this number of absolutely arbitrary parameters  is just too much for the fundamental theory.

Gravity is not fundamentally unied with the other interactions in the Standard
Model, although it is possible to graft on classical general relativity by
hand. However, general relativity is not a quantum theory, and there is no
obvious way to generate one within the standard model context.
Possible solutions include supergravity theories \DUrole{raw-tex}{\cite{Nilles:1983ge}}.
In addition to the fact that gravity is not unied and not quantized
there is another dificulty, namely the non-zero cosmological constant \DUrole{raw-tex}{$\Lambda_{cosm}$}.  The \emph{fine-tunning} of
primordial cosmological constant \DUrole{raw-tex}{$\Lambda_{bare}$},
\DUrole{raw-tex}{$$\Lambda_{cosm}=\Lambda_{bare} + \Lambda_{SSB},$$} \DUrole{raw-tex}{$$ \Lambda_{SSB}=8{\pi}G_N<0|V|0>\sim 10^{56}\Lambda_{obs} $$}
which can be thought of as the value of the energy of the vacuum in the absence of
spontaneous symmetry breaking, is required if someone  couple the electrowek interaction to the gravity.

Another question is: can all the interactions be unified?
Radiative effects make gauge couplings dependent on the energy scale.
The couplings, when defined as renormalized values including loop corrections, require the
specification of a renormalization prescription for which the modified minimal subtraction (MS)
scheme \DUrole{raw-tex}{\cite{tHooft:1973mm},\cite{Bardeen:1978yd}} is used.
In the SM the strong and weak couplings associated with non-Abelian gauge groups decrease
with energy, as it's shown for \DUrole{raw-tex}{$\alpha_S$} of strong interaction
in Figure \DUrole{raw-tex}{\ref{fig1}}, while the electromagnetic one associated with the Abelian group on the contrary
increases. Thus, it becomes possible that at some energy scale they become equal. According
to the Grand Unification Theory's (GUT) idea, this equality is not occasional but is a manifestation of a unique origin of
these three interactions. As a result of spontaneous symmetry breaking, the unifying group is
broken and the unique interaction is splitted into three branches which are called strong, weak and
electromagnetic interactions. Figure \DUrole{raw-tex}{\ref{fig2}} (left plot) clearly demonstrates that within the SM the coupling constant unification
at a single point is impossible. This result means that the unification can only be obtained if new physics enters between
the electroweak and the Planck scales, \DUrole{raw-tex}{$M_{P}=\sqrt{\hbar c^5/G_N} \sim 10^{19}$} GeV where Newton constant \DUrole{raw-tex}{$G_N$} is
extremely small \DUrole{raw-tex}{$G_N \sim 10 ^{-38}$} GeV\textsuperscript{-2}.  It turns out
that within the SUSY model a perfect unification can be obtained if the SUSY
masses are of an order of 1 TeV. This is shown in Figure \DUrole{raw-tex}{$\ref{fig2}$} (right plot).
From the fit requiring unification, one finds values, shown in Equation \DUrole{raw-tex}{\ref{eq1}}
for the break point \DUrole{raw-tex}{$M_{SUSY}$} and the unification point \DUrole{raw-tex}{$M_{GUT}$}. The unification is too
perfect in a supersymmetric theories.

\begin{flalign}\label{eq1}
M_{SUSY}   = 10^{3.4\pm 0.9\pm 0.4} GeV, \nonumber \\
M_{GUT}  = 10^{15.8\pm 0.3\pm 0.1} GeV, \nonumber \\
\alpha^{-1}_{GUT}  = 26.3 \pm 1.9 \pm 1.0
\end{flalign}
\begin{figure}
\noindent\makebox[\textwidth][c]{\includegraphics[width=0.800\linewidth]{/mnt/WorkingPlace/Thesis/thesis/theory/susy/plots/running_alphaS.PNG}}
\caption{Summary of running of the strong coupling \DUrole{raw-tex}{$\alpha_{S}$\cite{Bethke:2000ai}\label{fig1}}}
\end{figure}
\begin{figure}
\noindent\makebox[\textwidth][c]{\includegraphics[width=0.800\linewidth]{/mnt/WorkingPlace/Thesis/thesis/theory/susy/plots/GUT-SM-MSSM-3couplings.PNG}}
\caption{Evolution of the inverse of the three coupling constants in the Standard Model (left) and in the supersymmetric extension of the SM (MSSM) (right). Only in the latter case
unification is obtained. The SUSY particles are assumed to contribute only above the effective SUSY scale \DUrole{raw-tex}{$M_{SUSY}$} of about 1 TeV, which causes a change in the slope in the evolution of couplings. The thickness of the lines represents the error in the coupling constants. \DUrole{raw-tex}{\label{fig2}}}
\end{figure}

\DUrole{raw-tex}{\noindent}
Another solution for GUT is \DUrole{raw-tex}{SU(5)}. In the minimal \DUrole{raw-tex}{SU(5)} model \DUrole{raw-tex}{\cite{1974PhRvL..32..438G}}, interactions almost unify at
\DUrole{raw-tex}{$10^{16}GeV$}.  It's the simplest group which
has rank 4 (dimension of Cartan sub-algebra, SM has rank 4 as well ).
The matter fields (fermions and leptons) can be fitted well to  \DUrole{raw-tex}{$\bf\bar{5}$}  fundamental represanation and
\DUrole{raw-tex}{$\bf\bar{10}$}  representation by \DUrole{raw-tex}{5x5} antisymmetric matrices. Altogether, these are 15 degrees of freedom, just like in
one generation of SM. The unification within \DUrole{raw-tex}{SU(5)} results in only one coupling, quantization of the electric
charge \emph{Q} (the hypercharge is now a traceless generator \DUrole{raw-tex}{$Y$},   \DUrole{raw-tex}{$TrY=0$}), non-conserving barion and lepton numbers which
allows decays of protons with \DUrole{raw-tex}{$\tau \sim 10^{30} - 10^{31}$\cite{1978NuPhB.135...66B}}

To get the desired spontaneous symmetry breaking pattern in GUT, one needs two different scales \DUrole{raw-tex}{$V$}
and  \DUrole{raw-tex}{$v$} in a GUT , namely, \DUrole{raw-tex}{$M_W$} and \DUrole{raw-tex}{$M_{GUT}$} ,what leads
to a very serious problem which is called the hierarchy problem.

\begin{flalign}\label{eq2}
M_{H} \sim M_{W} \sim v \sim 10^2\, GeV, \nonumber \\
M_{GUT} \sim M_{\Xi}\sim V \sim 10^{16}\, GeV, \nonumber \\
M_{W}/M_{GUT}\sim 10^{-14},
\end{flalign}

\DUrole{raw-tex}{\noindent}
where \DUrole{raw-tex}{$H$} and \DUrole{raw-tex}{$\Xi$} are the scalar fields (light and heavy Higgs bosons )
responsible for the spontaneous breaking of SU(2) and GUT groups, respectively.
The question arises of how to get so small number \DUrole{raw-tex}{$M_{W}/M_{GUT}$} in a natural way. One needs some kind
of fine tuning in a theory, and we don’t know if there anything behind it.

Another aspect of the hierarchy problem is  the preservation of a given
hierarchy by \DUrole{raw-tex}{(\ref{eq2})}. The radiative corrections will destroy it.
To see this, consider the radiative correction to the light Higgs boson mass \DUrole{raw-tex}{$M_{H}$}. It is given by the Feynman
diagrams shown in Figure \DUrole{raw-tex}{\ref{fig3}}. The diagrams give the  ultra-violet (UV) quadratic divergency \DUrole{raw-tex}{(\ref{eq3})} at most.
\begin{figure}
\noindent\makebox[\textwidth][c]{\includegraphics[width=0.800\linewidth]{/mnt/WorkingPlace/Thesis/thesis/theory/susy/plots/Feynmann_comb.pdf}}
\caption{Tadpole and self-energy Feynmann diagrams for Higgs boson \DUrole{raw-tex}{\label{fig3}}. Here \DUrole{raw-tex}{$V$} stands for gauge bosons, \DUrole{raw-tex}{$e_k$} denotes
fermions. \DUrole{raw-tex}{\label{fig3}}}
\end{figure}

\begin{flalign}\label{eq3}
\delta M_{H}^2 \sim \lambda^2 \int \frac{d^4k}{(2\pi)^4}\frac{i}{k^2-M_H^2} ~ \lambda^2 \Lambda^2
    \end{flalign}

\DUrole{raw-tex}{\noindent}
where \DUrole{raw-tex}{$\lambda$} is a Yukawa coupling, \DUrole{raw-tex}{$\Lambda$} is  cut-off scale much larger than \DUrole{raw-tex}{$1\, TeV$},  perhaps of order of the Plank scale \DUrole{raw-tex}{$M_p$} or \DUrole{raw-tex}{$M_{GUT}$}.

The divergency spoils the hierarchy if it's not cancelled. This very accurate cancellation
with a precision \DUrole{raw-tex}{$\sim 10^{-14}$} needs a fine tuning of the coupling constants.
In the Standard Model, the  Higgs mass suffer from the quadratic divergent  correction, while the photon \DUrole{raw-tex}{$m_{\gamma}$} and
fermions \DUrole{raw-tex}{$m_f$} masses
are protected, because of  gauge  invariance and chiral symmetry, respectively.

\begin{flalign}\label{eq3}
\delta m_{\gamma}^2 \sim e^2 q^2 Log\Lambda^2,  \nonumber \\
\delta m_{f}^2 \sim e^2 m_f^2 Log\Lambda^2,
\end{flalign}

\DUrole{raw-tex}{\noindent}
where \DUrole{raw-tex}{$q$} is the photon momentum.

The only known way of achieving this kind of cancellation of quadratic terms (also known as
the cancellation of the quadratic divergencies) is supersymmetry (SUSY). Moreover, SUSY automatically
cancels quadratic corrections in all orders of Pertubation Theory (PT). This is due to the contributions of superpartners
of ordinary particles. The superparners  are originated   due to the invariance of the theory on supersymmetric
transformation. The contribution from boson loops cancels those from the fermion ones.
because of an additional factor (-1) coming from Fermi statistics, as shown in Figure \DUrole{raw-tex}{\ref{fig5}.}
\begin{figure}
\noindent\makebox[\textwidth][c]{\includegraphics[width=0.800\linewidth]{/mnt/WorkingPlace/Thesis/thesis/theory/susy/plots/Feynmann_SUSY_hierarcy_solving.pdf}}
\caption{Cancellation of quadratic divergency \DUrole{raw-tex}{\label{fig5}}.}
\end{figure}

The top diagrams, shown in Figure \DUrole{raw-tex}{\ref{fig5}}, are contributions of the Higgs bosons and its superpartner. The
strength is given by Yukawa coupling \DUrole{raw-tex}{$\lambda$}. The botom digrams of
Figure \DUrole{raw-tex}{\ref{fig5}} represents the gauge interaction of Higgs with gauge  bosons and gauginos
which is proportional to the gauge coupling \DUrole{raw-tex}{$g$}. The full cancellation takes place
in the case of unbroken supersymmetry when the sum rule \DUrole{raw-tex}{(\ref{eq4})} relating masses of bosons and their superpartners
is valid

\begin{flalign}\label{eq4}
\sum_{bosons} m^2 = \sum_{fermions} m^2
\end{flalign}

\DUrole{raw-tex}{\noindent}
If the equation \DUrole{raw-tex}{(\ref{eq4})} violated then SUSY is broken, and cancellation is true up to the SUSY breaking scale,
\DUrole{raw-tex}{$M_{SUSY}$}, given by \DUrole{raw-tex}{(\ref{eq5})}

\begin{flalign}\label{eq5}
\sum_{bosons} m^2 - \sum_{fermions} m^2  = M_{SUSY}^2
\end{flalign}

\DUrole{raw-tex}{\noindent}
\DUrole{raw-tex}{$M_{SUSY}$} should not be very large, \DUrole{raw-tex}{$M_{SUSY}\leq 1\, TeV$,} to make the \DUrole{raw-tex}{$fine-tuning$} natural, as a
consequence, the radiative correction will be of order of Higgs boson mass

\begin{flalign}\label{eq6}
\delta M_{H} \sim g^2 \cdot M_{SUSY}^2 \sim 10^{-1\cdot 2} \cdot 10^{3\cdot 2} \sim 10^{2\cdot 2} \sim M_H^2.
\end{flalign}


\section{2~~~Basics of supersymmetry%
  \label{basics-of-supersymmetry}%
}


\subsection{2.1~~~Superspace and Super-Poincare Lie Algebra%
  \label{superspace-and-super-poincare-lie-algebra}%
}

I start the introduction  to the supersymmetry by  explaining the superspace, superfileds  and
giving an  algebra of supersymmetric transformation,
\DUrole{raw-tex}{Super-Poincare Lie} Algebra, which contains additional SUSY generators \DUrole{raw-tex}{$Q_\alpha^i$} and \DUrole{raw-tex}{$\bar{Q}_{\alpha}^i$}.
Here  \DUrole{raw-tex}{$\alpha$} and \DUrole{raw-tex}{$i$} are  spinorial  and supersymmetric indexes. I'll illustrate the ideas of
supersymmetry using  a simple example having only one supersymmetric dimension \DUrole{raw-tex}{$i=1$} which is
usually  denoted as
SUSY with only one chiral supermultiplet, \DUrole{raw-tex}{$N=1$}. The model was first written by Wees and Zumino \DUrole{raw-tex}{\cite{Wess:1974jb}}
The spinorial SUSY charge \DUrole{raw-tex}{$Q_{\alpha}^i$} performs transformations of the matter fields,
when supertranslation in the superspace is done. The superspace \DUrole{raw-tex}{\cite{}} differs from the ordinary
Euclidean (Minkowski) space by  adding of two new coordinates, \DUrole{raw-tex}{$\theta_{\alpha}$} and \DUrole{raw-tex}{$\bar{\theta}_{\alpha}$}
which are Grassmanian, ie anticommuting, variables \DUrole{raw-tex}{($\ref{eq7}$)},
where \DUrole{raw-tex}{$\bar{\theta}$} variable is obtained from conjugating
\DUrole{raw-tex}{$\theta$}

\begin{flalign}\label{eq7}
\left\{\theta_{\alpha},\theta_{\beta}\right\} = \left\{\bar{\theta}_{\alpha},\bar{\theta}_{\beta}\right\}=0,     \nonumber \\
\left\{\frac{\partial}{\partial \theta_{\alpha}},\theta_{\beta}\right\} =  \left\{\frac{\partial}{\partial \bar{\theta}_{\alpha}},\bar{\theta}_{\beta}\right\} = \delta_{\alpha\beta}, \nonumber \\
\left\{\frac{\partial}{\partial \theta_{\alpha}},\bar{\theta}_{\beta}\right\} =  \left\{\frac{\partial}{\partial \bar{\theta}_{\alpha}},\theta_{\beta}\right\} = 0,
\end{flalign}

\DUrole{raw-tex}{\noindent}
The Minkowski space  transforms to superspace, as shown in \DUrole{raw-tex}{($\ref{eq8}$)}

\begin{flalign}\label{eq8}
{x_\mu} \rightarrow  {x_\mu,\theta,\bar{\theta}}
\end{flalign}

\DUrole{raw-tex}{\noindent}
A SUSY group element \DUrole{raw-tex}{$(\ref{eq9})$} in the representation applicable for Weyl spinors and scalar fields
can be constructed in  the superspace in the same way as an ordinary
translation in the usual space

\begin{flalign}\label{eq9}
G(x_\mu,\theta_{\alpha},\bar{\theta}_{\dot{\alpha}}) = e^i\cdot(-x^{\mu}P_{\mu}+\theta_{\alpha}Q_{\alpha}+\bar{\theta}_{\dot{\alpha}}\bar{Q}_{\dot{\alpha}})
\end{flalign}

\DUrole{raw-tex}{\noindent}
I'll skip spinorial indexes in further calculations assuming the following notation for the representations of
scalars, pseudoscalars and 4-vectors constructed on Grassman variables

\DUrole{raw-tex}{$$\theta\theta \rightarrow \epsilon^{\alpha\beta}\theta_{\alpha}\theta_{\beta}, \\
\bar{\theta}\bar{\theta} \rightarrow \epsilon^{\dot{\alpha}\dot{\beta}}\bar{\theta}_{\dot{\alpha}}\bar{\theta}_{\dot{\beta}}, \\
\bar{\theta}\sigma^{\mu}\theta \rightarrow \bar{\theta}^{\dot{\alpha}}\sigma^{\mu}_{\dot{\alpha}\alpha}\theta^{\alpha},
$$}

where \DUrole{raw-tex}{$\sigma^{\mu}$} \DUrole{raw-tex}{\cite{}} defined by  \DUrole{raw-tex}{$2\times2$} identity matrix \DUrole{raw-tex}{$I_{2\times2}$} and three vector of
Pauli matrices  \DUrole{raw-tex}{$\vec{\sigma}=(\sigma^1,\sigma^2,\sigma^3)$} \DUrole{raw-tex}{\cite{}} \DUrole{raw-tex}{$$\sigma^{\mu}=(I_{2x2},\vec{\sigma})$$}
and \DUrole{raw-tex}{$\epsilon^{\alpha\beta}$} is the antisymmetric tensor \DUrole{raw-tex}{\cite{}}
Any infinitesimal  trnasformations in the superspace induced by \DUrole{raw-tex}{$(\ref{eq9})$} take the form \DUrole{raw-tex}{$(\ref{eq10})$}

\begin{flalign}\label{eq10}
 x_{\mu} \rightarrow  x_{\mu} + i\theta\sigma_{\mu}\bar{\epsilon} + i\epsilon\sigma_{\mu}\bar{\theta}, \nonumber \\
 \theta \rightarrow  \theta + \epsilon, \nonumber \\
 \bar{\theta} \rightarrow  \bar{\theta} + \bar{\epsilon}
\end{flalign}

\DUrole{raw-tex}{\noindent}
Someone can deduce the expressions of \DUrole{raw-tex}{$Q$} and \DUrole{raw-tex}{$\bar{Q}$}  SUSY generators from \DUrole{raw-tex}{$(\ref{eq9})$} and \DUrole{raw-tex}{$(\ref{eq10})$}

 \begin{flalign}\label{eq11}
  Q = \frac{\partial}{\partial \theta} - i \bar{\theta}\sigma_{\mu}\partial_{\mu}, \nonumber \\
  \bar{Q} = -\frac{\partial}{\partial \bar{\theta}} + i \sigma_{\mu}\theta\partial_{\mu}
\end{flalign}

\DUrole{raw-tex}{\noindent}
It's obvious to show the main rule of the graded (Super-Poinkare )  Lie Algebra \DUrole{raw-tex}{\cite{}}. This is  the
anticommutator of  \DUrole{raw-tex}{$Q$} and \DUrole{raw-tex}{$\bar{Q}$}.

 \begin{flalign}\label{eq11}
 \{Q_\alpha,\bar{Q}_\beta\} = 2 \sigma^{\mu}_{\alpha\beta}P_{\mu}
\end{flalign}

\DUrole{raw-tex}{\noindent}
The presence of the translation generator in \DUrole{raw-tex}{$(\ref{eq11})$} shows that the supersymmetry is a spacetime
symmetry. It means that the supersymmetry is conserved in time \DUrole{raw-tex}{$(\ref{eq12})$}

\begin{flalign}\label{eq12}
[Q_{\alpha},P^0] = 0
\end{flalign}

\DUrole{raw-tex}{\noindent}
More general case of  the equtation \DUrole{raw-tex}{$(\ref{eq12})$}  can be written as \DUrole{raw-tex}{$(\ref{eq13})$}

\begin{flalign}\label{eq13}
[Q_{\alpha},P^{\mu}] = 0
\end{flalign}

\DUrole{raw-tex}{\noindent}
The equation \DUrole{raw-tex}{$(\ref{eq13})$}  is the second rule of the graded Lie Algebra.
The commutators of \DUrole{raw-tex}{$Q(\bar{Q})$} with the Lorentz group generators \DUrole{raw-tex}{$M_{\mu\nu}$} of  the angular
momentum are fixed because the supersymmetric charge was declared to be a spin \DUrole{raw-tex}{$1/2$} Weyl spinor, i.e \DUrole{raw-tex}{$Q(\bar{Q})\sim \psi_{L(R)}$}.
The commutator  becomes

\begin{flalign}\label{eq14}
[Q_\alpha,M^{\mu\nu}] = 1/2\sigma^{\mu\nu}_{\alpha\beta}Q_{\beta}=-[\bar{Q}_{\alpha},M_{\mu\nu}] , \nonumber\\
 \sigma^{\mu\nu}_{\alpha\beta}=\frac{i}{4}[\sigma_{\mu},\sigma_{\nu}]
\end{flalign}

\DUrole{raw-tex}{\noindent}
Models with more than one SUSY charge, \DUrole{raw-tex}{$Q_{\alpha}$} (extended SUSY) in the low energy theory do not lead to chiral fermions and so
are excluded for phenomenological reasons.

We immediately note that \DUrole{raw-tex}{$(\ref{eq13})$} implies that the zero energy
state (the vacuum)  comes in degenerate pairs of the states having the same energy, where one member is
a boson and the other one is  a fermion.
This is shown in the  equations \DUrole{raw-tex}{$(\ref{eq15a})$} :

\begin{flalign}\label{eq15a}
 |0> = |E,\lambda>,\, E=0, \nonumber \\
 E=1/4Tr<0|\{Q_{\alpha},\bar{Q}_{\alpha}\}|0>=1/4|Q_{\alpha}|0>|^2 = 0, \,\, if \nonumber \\
  Q_{\alpha}|0>=0,\,\,\,\bar{Q}_{\alpha}|0>=|E,\lambda+1/2>,\,\,\bar{Q}_{\alpha}^i\bar{Q}_\beta^j|0>=|E,\lambda+1> \,\,\, etc
\end{flalign}

\DUrole{raw-tex}{\noindent}
This the degeneracy of the vacum  is destroyed,  if the invariance of the   vacum  is spontaneously broken, i.e
\DUrole{raw-tex}{$$ Q_{\alpha}|0>=|E,\lambda-1/2> \neq 0, \,\, E>0,\,\, \bar{Q}_{\alpha}|0>=|E^{\prime},\lambda+1/2>,\,\, E\neq E^{\prime} $$}
Here \DUrole{raw-tex}{$\lambda$} is a helicity of ground state. The \DUrole{raw-tex}{$N-$} supersymmetry considers  the vacum as the  multipartilce state
\DUrole{raw-tex}{$$\bar{Q}_1\bar{Q}_2\bar{Q}_3...\bar{Q}_N|0>=|E,\lambda+N/2>$$}.  Thus number of supersymmetries \DUrole{raw-tex}{$N$}
relates to the maximal spin of the particle \DUrole{raw-tex}{$S$}  in the multiplet as \DUrole{raw-tex}{$$ -S+N/2\leq S \Rightarrow N\leq 4S. $$}
Here  multiplet is assumed to be invariant under CPT transformation, i.e any state is doubly degenerated
\DUrole{raw-tex}{$|E,\pm\lamda(\pm 1/2;\pm 1; etc)>$}.
The total number of the possible states in
the \DUrole{raw-tex}{$N-$} supersymmetry theory is \DUrole{raw-tex}{$2^N=2^{N-1}\,\,bosons+2^{N-1}\,\,fermions$}
The quantum field theories with \DUrole{raw-tex}{$S>1$} are non-renormalizable. Hence I will restrict the consideration of SUSY to \DUrole{raw-tex}{$N =  1$}
which contains two types of the supermultiplets \DUrole{raw-tex}{$(\ref{eq15b})$}: chiral and vector multiplets

\begin{flalign}\label{eq15b}
 Scalar=\phi = |E,\lambda= 0>,\, Fermion=\xi=\bar{Q}|\phi> = |E,\lambda=1/2>,   \nonumber \\
 Fermion=\xi=|E,\lambda= 1/2>,\, Vector=A_{\mu}=\bar{Q}|\xi> = |E,\lambda=1>
\end{flalign}

\DUrole{raw-tex}{\noindent}
Thus in any supersymmetric theory,
every particle has a partner with the same mass but with a spin differing by \DUrole{raw-tex}{$1/2$}
(since \DUrole{raw-tex}{$Q$} is  a spinorial operator)
SUSY acts independently of any internal symmetry. In other words, the
generators of supersymmetry commute with all internal symmetry generators.
As a result, any particle and its superpartner have identical
internal quantum numbers such as electric charge, isospin, colour, etc.


\subsection{2.2~~~Superfields.%
  \label{superfields}%
}

A construction of SUSY invariant Lagrangiants requires to introduce chiral and vector superfields. Their
formalism  can  easily be used to write out the general SUSY invariant  Lagrangian with local gauge
interaction.
In general, the superfield is an anlytic function \DUrole{raw-tex}{$\mathcal{F}(z)$} defined in the superspace \DUrole{raw-tex}{$z$}. The function contains terms which
are proportional Grassmanians in some power of \DUrole{raw-tex}{$\theta,\bar{\theta}$} up to 4 at most: \DUrole{raw-tex}{$1,\theta, \bar{\theta},\theta\theta, \bar{\theta}\bar{\theta},
\theta\theta\bar{\theta}, etc$} . The most general view of the superfield is

\begin{flalign}\label{eq166}
\mathcal{F}(z)\equiv \mathcal{F}(x,\ \theta,\overline{\theta})\ =\ f(x)+\sqrt{2}\theta\xi(x)+\sqrt{2}\overline{\theta}\overline{\chi}(x)+\theta\theta M(x)+\overline{\theta}\overline{\theta}N(x)+ \nonumber \\
\theta\sigma^{\mu}\overline{\theta}A_{\mu}(x)+\theta\theta\overline{\theta}\overline{\lambda}(x)+\overline{\theta}\overline{\theta}\theta\zeta(x)+\frac{1}{2}\theta\theta\overline{\theta}\overline{\theta}D(x)
 \end{flalign}

\DUrole{raw-tex}{\noindent}
This is the most general form since \DUrole{raw-tex}{$\overline{\theta}\overline{\sigma}^{\mu}\theta=-\theta\sigma^{\mu}\overline{\theta}$}
and \DUrole{raw-tex}{$\theta\sigma^{\mu\nu}\theta=0=\overline{\theta}\overline{\sigma}^{\mu\nu}\overline{\theta}$}.
The \DUrole{raw-tex}{$\sqrt{2}$} coefficients have been
put in for convenience with respect to supersymmetry transformations.
One can consider the hermitian conjugate \DUrole{raw-tex}{$\mathcal{F}^{\dagger}(z)$} as
an independent superfield.
\DUrole{raw-tex}{$\mathcal{F}(z)$} doesn't need to have a well-defined parity. There are  several component fields of
the superfield \DUrole{raw-tex}{$\mathcal{F}(x,\ \theta,\overline{\theta})$}. They are
scalar fields \DUrole{raw-tex}{$f(x),\ M(x),\ N(x)$} and \DUrole{raw-tex}{$D(x)$}, one vector field \DUrole{raw-tex}{$A_{\mu}(x)$} plus two left handed Weyl spinor
fields \$xi\_\{A\}(x),zeta\_\{A\}(x)\$ and two right handed Weyl spinor fields
\DUrole{raw-tex}{$\overline{\chi}^{A}(x)$} and \DUrole{raw-tex}{$\overline{\lambda}^{\dot{A}}(x)$}.
All fields being complex, there are sixteen real bosonic and sixteen real fermionic component fields.
These are, of course, off-shell degrees of freedom since we have not imposed the equations of motion.
They  correspond (though not necessarily in a one to one way) to particles that neatly fall into complete supermultiplets.
The effect of the infinitesimal supersymmetry transformation \DUrole{raw-tex}{$\ref{eq10}$} on the components of \DUrole{raw-tex}{$\mathcal{F}$}
can be worked out \DUrole{raw-tex}{\cite{Salam:1974yz},\cite{Salam:1974jj},\cite{Wess:1974jb}}
from the requirement that \DUrole{raw-tex}{$\delta \mathcal{F}$} has the same form as  the superfield \DUrole{raw-tex}{$(\ref{eq166})$}. Thus
the variation of components obey \DUrole{raw-tex}{$(\ref{eq16})$}

\begin{flalign}\label{eq16}
\delta \mathcal{F}\equiv  \mathcal{F}(x^{\mu}-i\theta\sigma^{\mu}\overline{\epsilon}+
i\epsilon\sigma^{\mu}\overline{\theta},\ \theta+\epsilon,\overline{\theta}+
\overline{\epsilon})-\mathcal{F}(x,\ \theta,\overline{\theta})=i(\epsilon Q+\overline{\epsilon}\overline{Q})\mathcal{F},\nonumber \\
\delta f=\sqrt{2}\epsilon\xi+\sqrt{2}\overline{\epsilon}\overline{\chi}, \nonumber \\
\delta(\sqrt{2}\xi_{A})=2\epsilon_{A}M+(\sigma^{\mu}\overline{\epsilon})_{A}(-i\partial_{\mu}f+A_{\mu}), \nonumber \\
\delta(\sqrt{2}\overline{\chi}^{\dot{A}})=2\overline{\epsilon}^{A}N-(\overline{\sigma}^{\mu}\epsilon)^{A}(i\partial_{\mu}f+A_{\mu}), \nonumber \\
\displaystyle \delta M=\overline{\epsilon}\overline{\lambda}+\frac{i}{\sqrt{2}}\partial_{\mu}\xi\sigma^{\mu}\overline{\epsilon}, \nonumber \\
\displaystyle \delta N=\epsilon\zeta-\frac{i}{\sqrt{2}}\epsilon\sigma^{\mu}\partial_{\mu}\overline{\chi}, \nonumber \\
\delta \mathrm{A}_{\mu}\ =\ \epsilon\sigma_{\mu}\overline{\lambda}+\zeta\sigma_{\mu}\overline{\epsilon}-\frac{i}{\sqrt{2}}\epsilon\partial_{\mu}\xi+\frac{i}{\sqrt{2}}\partial_{\mu}\overline{\chi}\overline{\epsilon}, \nonumber  \\
+\sqrt{2}\epsilon\sigma_{\mu\nu}\partial^{\nu}\xi-\sqrt{2}\overline{\epsilon}\overline{\sigma}_{\mu\nu}\partial^{\nu}\overline{\chi},   \nonumber \\
\displaystyle \delta\overline{\lambda}^{A}=\overline{\epsilon}^{A}D-\frac{i}{2}\overline{\epsilon}^{A}\partial^{\mu}A_{\mu}-i(\overline{\sigma}^{\mu}\epsilon)^{A}\partial_{\mu}M+(\overline{\sigma}^{\mu\nu}\overline{\epsilon})^{A}\partial_{\mu}A_{\nu}, \nonumber \\
\displaystyle \delta\zeta_{A}=\epsilon_{A}D+\frac{i}{2}\epsilon_{A}\partial^{\mu}A_{\mu}-i(\sigma^{\mu}\overline{\epsilon})_{A}\partial_{\mu}N-(\sigma^{\mu\nu}\epsilon)_{A}\partial_{\mu}A_{\nu}, \nonumber \\
\delta D=i\partial_{\mu}(\zeta\sigma^{\mu}\overline{\epsilon}+\overline{\lambda}\overline{\sigma}^{\mu}\epsilon)
\end{flalign}

\DUrole{raw-tex}{\noindent}
\DUrole{raw-tex}{$\delta D$} being a four divergence, with the usual assumption of discarding surface terms,
(i.e. coefficient of \DUrole{raw-tex}{$\theta\theta\overline{\theta}\overline{\theta}$} in a superfield)
in the Lagrangian density would yield a supersymmetric action.
Linear combinations of superfields are again superfields (since \DUrole{raw-tex}{$Q,\overline{Q}$} are linear differential operators),
i.e. superfields form linear representations of the supersymmetry algebra.
Products of superfields will be general superfields.
Some of the component fields (called auxiliary fields, for example \DUrole{raw-tex}{$M,N,D$}) do not contribute to an on-shell description,
and can be ruled out by impliying the equation of motion.
Superfield representations are highly reducible.
i.e. are physically redundant. It's possible to obtain irreducible representaion if someone introduce additional constraints
on  the supermultiplet.
Specific types of irreducible superfields are
chiral and vector superfields. However, products of irreducible superfields may or may not be irreducible superfields.


\subsubsection{2.2.1~~~Chiral superfields.%
  \label{chiral-superfields}%
}

I'm going to introduce chiral and vector supermultilet as building blocks of the SUSY invariant Lagrangian.
First, left and right chiral covariant derivatives \DUrole{raw-tex}{$\partial_A$} will be considered \DUrole{raw-tex}{\cite{Wess:1974jb}}. The spinorial derivative is not conserved
under the supersymmetric transformation
\DUrole{raw-tex}{$x^{\prime\mu}\equiv x^{\mu}-i\theta\sigma^{\mu}\overline{\epsilon}+i\epsilon\sigma^{\mu}\overline{\theta},\ \theta'\equiv\theta+\epsilon,\overline{\theta}'\equiv\overline{\theta}+\overline{\epsilon}$}
\DUrole{raw-tex}{$$ \partial_{A}=\frac{\partial\theta^{\prime B}}{\partial\theta^{A}}\frac{\partial}{\partial\theta^{\prime B}}+\frac{\partial x^{\prime\mu}}{\partial\theta^{A}}\frac{\partial}{\partial x^{\prime\mu}}
=\frac{\partial}{\partial\theta^{A}}-i(\sigma^{\mu}\overline{\epsilon})_{A}\frac{\partial}{\partial x^{\mu}} \ \ \
\frac{\partial}{\partial^{\prime}_A} \neq   \frac{\partial}{\partial_A}
$$}. Also \DUrole{raw-tex}{$\partial_A$} doesn't commute with the supersymmetry charges \DUrole{raw-tex}{$Q$}, \DUrole{raw-tex}{$ \bar{Q}$} in
contradiction to the momentum operator \DUrole{raw-tex}{$\partial_{\mu}=-iP_{\mu}$}. Thus the  left and right chiral covariant derivatives
defined as   \DUrole{raw-tex}{$(\ref{eq17})$}

\begin{flalign}\label{eq17}
 D_{A}\equiv\partial_{A}-i\sigma_{AB}^{\mu}\overline{\theta}^{B}\partial_{\mu}, \nonumber \\
 \overline{D}_{A}\equiv -\overline{\partial}_{A}+i\theta^{B}\sigma_{BA}^{\mu}\partial_{\mu}
\end{flalign}

\DUrole{raw-tex}{\noindent}
anticommutes with  \DUrole{raw-tex}{$Q$}, \DUrole{raw-tex}{$\bar{Q}$} operators    as shown in \DUrole{raw-tex}{$(\ref{eq18})$}

\begin{flalign}\label{eq18}
\left\{ D_{A},\bar{Q}_B\right\} = \left\{\bar{D}_A,Q_B \right\}=0, \nonumber \\
\left\{ \bar{D}_{A},\bar{Q}_B\right\} = \left\{D_A,Q_B \right\}=0
\end{flalign}

\DUrole{raw-tex}{\noindent}
Also, it follows from \DUrole{raw-tex}{$(\ref{eq18})$}  that \DUrole{raw-tex}{$D_A(\bar{D_A})$} is invariant under supersymmetric transformation \DUrole{raw-tex}{$\delta =
i(\epsilon Q + \bar{\epsilon}\bar{Q})$}

\begin{flalign}\label{eq19}
  D_A (x^{\prime},\theta^{\prime},\bar{\theta}^{\prime} )  = D_A (x,\theta,\bar{\theta} ), \nonumber \\
 \bar{D}_A (x^{\prime},\theta^{\prime},\bar{\theta}^{\prime} ) = \bar{D}_A (x,\theta,\bar{\theta} ), \nonumber \\
 \delta (\bar{D}_A \Phi(z)) = \delta (\bar{D}_A) \Phi(z) + \bar{D}_A  \delta (\Phi(z)) = \bar{D}_A  \delta (\Phi(z))
\end{flaign}

\DUrole{raw-tex}{\noindent}
If someone performs the shift \DUrole{raw-tex}{$y^{\mu}=x^{\mu}-i\theta\sigma^{\mu}\bar{\theta}$} or \DUrole{raw-tex}{$\overline{y}^{\mu}
\equiv x^{\mu}+i\theta\sigma^{\mu}\overline{\theta}$} in superspace \DUrole{raw-tex}{$z,$} then the
right (left) chiral covariant derivative \DUrole{raw-tex}{$\bar{D}_A$} (\DUrole{raw-tex}{$D_A$}) is zero for
any analytic function \DUrole{raw-tex}{$f(y,\theta)\,\,(f^{\star}(\bar{y},\bar{\theta}))$} in the point \DUrole{raw-tex}{$y(\overline{y})$}

\begin{flalign}\label{eq20}
\overline{D}_{A} y^{\mu}=0, D_{A}\overline{y}^{\mu}=0, \nonumber \\
\overline{D}_{A} f(y,\ \theta)= 0, D_{A}f^{\star}(\overline{y},\overline{\theta})=0
\end{flalign}

\DUrole{raw-tex}{\noindent}
A multiplet \DUrole{raw-tex}{$\Phi(y,\theta)\,\,(\Phi^{+}(\bar{y},\bar{\theta}))$} is left (right) chiral superfield,
if it  obeys the equation \DUrole{raw-tex}{$(\ref{eq21})$}

\begin{flalign}\label{eq21}
 \bar{D}\Phi (y,\theta) =0, \nonumber \\
 D \Phi^{\dagger} (\bar{y},\bar{\theta}) =0
\end{flalign}

\DUrole{raw-tex}{\noindent}
The decomposition of these superfields in terms of their component fields
can be given as

\begin{flalign}\label{eq21}
 \Phi(y,\ \theta)=\phi(y)+\sqrt{2}\theta\xi(y)+\theta\theta F(y),\nonumber \\
 \Phi^{\dagger}(\overline{y}_{)}\overline{\theta})=
 \phi^{\star}(\overline{y})+\sqrt{2}\overline{\theta}\overline{\xi}(\overline{y})+
 \overline{\theta}\overline{\theta}F^{\star}(\overline{y})
\end{flalign}

\DUrole{raw-tex}{\noindent}
Taylor expansion of \DUrole{raw-tex}{(\ref{eq21})}  using substitutions \DUrole{raw-tex}{$y^{\mu}=x^{\mu}-i\theta\sigma^{\mu}\overline{\theta}$}, and
\DUrole{raw-tex}{$\overline{y}^{\mu}=x^{\mu}+i\theta\sigma^{\mu}\overline{\theta}$} gives us

\begin{flalign}\label{eq22}
\Phi(y,\ \theta)\ =\ \phi(x)-i\theta\sigma^{\mu}\overline{\theta}\partial_{\mu}\phi(x)-
\frac{1}{4}\theta\theta\overline{\theta}\overline{\theta}\partial^{\mu}\partial_{\mu}\phi(x)+\sqrt{2}\theta\xi(x), \nonumber \\
 +\displaystyle \frac{i}{\sqrt{2}}\theta\theta\partial_{\mu}\xi\sigma^{\mu}\overline{\theta}+\theta\theta F(x),  \nonumber \\
 \Phi^{\dagger}(\overline{y},\overline{\theta})\ =\ \phi^{\star}(x)+i\theta\sigma^{\mu}\overline{\theta}\partial_{\mu}\phi^{\star}(x)-\frac{1}{4}\theta\theta\overline{\theta}
 \overline{\theta}\partial^{\mu}\partial_{\mu}\phi^{\star}(x)+\sqrt{2}\overline{\theta}\overline{\xi}(x) \nonumber \\
 -\displaystyle \frac{i}{\sqrt{2}}\overline{\theta}\overline{\theta}\theta\sigma^{\mu}\partial_{\mu}\overline{\xi}(x)+\overline{\theta}\overline{\theta}F^{\star}(x)
\end{flalign}

\DUrole{raw-tex}{\noindent}
Comparing \DUrole{raw-tex}{$(\ref{eq22})$} with \DUrole{raw-tex}{$(\ref{eq16})$}, the supersymmetric transformations of the chiral components can be expressed as

\begin{flalign}\label{eq23}
\delta\phi=\sqrt{2}\epsilon\xi, \nonumber  \\
\delta\xi_{A}=\sqrt{2}\epsilon_{A}F-\sqrt{2}i(\sigma^{\mu}\overline{\epsilon})_{A}\partial_{\mu}\phi, \nonumber \\
\delta F=i\partial_{\mu}(\sqrt{2}\xi\sigma^{\mu}\overline{\epsilon})
\end{flalign}

\DUrole{raw-tex}{\noindent}
Component fields in \DUrole{raw-tex}{$\Phi\dagger$} obey the corresponding hermitian conjugate transformations.
The \DUrole{raw-tex}{$F$}-component of \DUrole{raw-tex}{$\Phi$} transforms into itself plus a spacetime derivative.
Hence, such a term in the Lagrangian density, called an F-term,
leads to a supersymmetry invariant action when surface terms can be discarded.
Products of chiral superfields \DUrole{raw-tex}{$\Phi_{1}\Phi_{2}\cdots\Phi_{l}$} or \DUrole{raw-tex}{$\Phi_{1}^{\dagger}\Phi_{2}^{\dagger}
\cdots\Phi_{l}^{\dagger}$} are also chiral superfields themselves as shown in \DUrole{raw-tex}{$(\ref{eq24})$}

\begin{flalign}\label{eq24}
\Phi_{i}\Phi_{j}=\phi_{i}\phi_{j}+\sqrt{2}\theta(\xi_{i}\phi_{j}+\phi_{i}\xi_{j})+\theta\theta(\phi_{i}F_{j}+\phi_{j}F_{i}-\xi_{i}\xi_{j}), \nonumber \\
\Phi_{i}\Phi_{j}\Phi_{k}\ =\ \phi_{i}\phi_{j}\phi_{k}+\sqrt{2}\theta(\xi_{i}\phi_{j}\phi_{k}+\xi_{j}\phi_{k}\phi_{i}+\xi_{k}\phi_{i}\phi_{j}) \nonumber \\
+\theta\theta(F_{i}\phi_{j}\phi_{k}+F_{j}\phi_{k}\phi_{i}+F_{k}\phi_{i}\phi_{j}-\xi_{i}\xi_{j}\phi_{k}-\xi_{j}\xi_{k}\phi_{i}-\xi_{k}\xi_{i}\phi_{j}), \nonumber \\
\bar{D}(\Phi_{i}\Phi_{j})= \bar{D} (\Phi_{i}\Phi_{j}\Phi_{k}) = 0
\end{flalign}

\DUrole{raw-tex}{\noindent}
The important point to note is that in the Lagrangian density
the \DUrole{raw-tex}{$F$}-term in any polynomial function of chiral superfields would yield a supersymmetric action
\DUrole{raw-tex}{$S=\displaystyle \int d^4xd^2\theta\Phi_{1}(z)\cdots\Phi_{n}(z),$} where  the Lagrangian density
\DUrole{raw-tex}{$\mathcal{L}$} is \DUrole{raw-tex}{$\Phi_{1}(z)\cdots\Phi_{n}(z) |_{\theta\theta}$} (\DUrole{raw-tex}{$ |_{\theta\theta}$} denotes \DUrole{raw-tex}{$F$}-term).
Such is not the case for the product of a chiral superfield with
its hermitian conjugate \DUrole{raw-tex}{$\Phi_{i}^{\dagger}\Phi_{j}$}. The product is a vector superfield which \DUrole{raw-tex}{$D$}-component,
coefficient of \DUrole{raw-tex}{$\theta\theta\overline{\theta}\overline{\theta}$}
(\DUrole{raw-tex}{$ |_{\theta\theta\overline{\theta}\overline{\theta}}$}), appears in a Lagrangian density yeilding a supersymmetric
action \DUrole{raw-tex}{$S=\int d^{4}xd^2\theta d^2\bar{\theta}\Phi_{i}^{\dagger}\Phi_{j}=\int d^4 x \Phi_{i}^{\dagger}\Phi_{j}
|_{\theta\theta\overline{\theta}\overline{\theta}}$}.


\subsubsection{2.2.2~~~Vector superfields.%
  \label{vector-superfields}%
}

A vector superfield  requires reality \DUrole{raw-tex}{$$ V = V^{\dagger}. $$}
Having in mind the general form of superfield \DUrole{raw-tex}{$(\ref{eq16})$}, the vector superfield \DUrole{raw-tex}{$V(x,\theta\bar{\theta})$}
is expressed as the following

\begin{flalign}\label{eq25}
 V(x,\ \theta,\overline{\theta})\ \sim\ C(x)+\sqrt{2}\theta\xi(x)+\sqrt{2}\overline{\theta}
 \overline{\xi}(x)+\theta\theta M(x)+\overline{\theta}\overline{\theta}M^{\star}(x)+
 \theta\sigma^{\mu}\overline{\theta}A_{\mu}(x) \nonumber \\
 +\theta\theta\overline{\theta}\overline{\lambda}(x)+\overline{\theta}\overline{\theta}
  \theta\lambda(x)+\frac{1}{2}\theta\theta\overline{\theta}\overline{\theta}D(x),
\end{flalign}

\DUrole{raw-tex}{\noindent} where \DUrole{raw-tex}{$C(x),\ A_{\mu}(x)$} and \DUrole{raw-tex}{$D(x)$} are real fields, \DUrole{raw-tex}{$M(x)$} is a complex scalar field
and \DUrole{raw-tex}{$\xi(x),\ \lambda(x)$} are complex two component spinor fields. Vector superfields can be constructed
from a chiral superfield \DUrole{raw-tex}{$\Phi$} in the way like \DUrole{raw-tex}{$$ V\sim \Phi + \Phi^{\dagger}, \ \  \sim \Phi^{\dagger}\Phi $$}
Using \DUrole{raw-tex}{$(\ref{eq22})$}, and replacing \DUrole{raw-tex}{$\xi(x)$} by \DUrole{raw-tex}{$\chi(x)$}, we can write

 \begin{flalign}\label{eq26}
  \Phi + \Phi^{\dagger} = 2 \Re \phi(x)+\sqrt{2}\theta\chi(x)+ \sqrt{2}\overline{\theta}
 \overline{\chi}(x)+\theta\theta F(x)+\overline{\theta}\overline{\theta}F^{\star}(x) \nonumber \\
+2\theta\sigma^{\mu}\overline{\theta}\partial_{\mu}\Im\phi(x)-\frac{i}{\sqrt{2}}
\theta\theta\overline{\theta}\overline{\sigma}^{\mu}\partial_{\mu}\chi(x)-\frac{i}{\sqrt{2}}
\overline{\theta}\overline{\theta}\theta\sigma^{\mu}\partial_{\mu}\overline{\chi}(x) \nonumber \\
- \frac{1}{2}\theta\theta\overline{\theta}\overline{\theta}\partial^{\mu}\partial_{\mu}\Re\phi(x)
 \end{flalign}

\DUrole{raw-tex}{\noindent}
The definition \DUrole{raw-tex}{$(\ref{eq25})$} can be  rewritten  with addtitional substitutions \DUrole{raw-tex}{$\lambda-i\sigma^{\mu}\partial_{\mu}
\overline{\xi}/\sqrt{2}$} for
\DUrole{raw-tex}{$\lambda,\ \overline{\lambda}-i\overline{\sigma}^{\mu}\partial_{\mu}\xi/\sqrt{2}$} for \DUrole{raw-tex}{$\overline{\lambda}$} and
\DUrole{raw-tex}{$D-\displaystyle \frac{1}{2}\partial^{\mu}\partial_{\mu}C$} for \DUrole{raw-tex}{$D$}. Thus a more general vector superfield  defined as

\begin{flalign}\label{eq27}
V(z)\ =\ C(x)+\sqrt{2}\theta\xi(x)+\sqrt{2}\overline{\theta}\overline{\xi}(x)+\theta\theta M(x)+\overline{\theta}\overline{\theta}M^{\star}(x)+\theta\sigma^{\mu}\overline{\theta}\mathrm{A}_{\mu}(x) \nonumber \\
+\theta\theta\overline{\theta}\{\overline{\lambda}(x)-\frac{i}{\sqrt{2}}\overline{\sigma}^{\mu}\partial_{\mu}\xi(x)\}+\overline{\theta}\overline{\theta}\theta\{\lambda(x)-\frac{i}{\sqrt{2}}\sigma^{\mu}\partial_{\mu}\overline{\xi}(x)\} \nonumber \\
+ \frac{1}{2}\theta\theta\overline{\theta}\overline{\theta}\{D(x)-\frac{1}{2}\partial^{\mu}\partial_{\mu}C(x)\}
\end{flalign}

\DUrole{raw-tex}{\noindent}
can be \emph{supergauge transformed} with changed field components \DUrole{raw-tex}{$ C\rightarrow C+2\Re e\phi,\ \xi\rightarrow\xi+\chi,\ M\rightarrow M+F,\
A_{\mu}\rightarrow A_{\mu}-2\partial_{\mu}\Im m\phi,\ \lambda\rightarrow\lambda,\ D\rightarrow D$}. There is a particular gauge,
\emph{Wess-Zummino gauge} \DUrole{raw-tex}{\cite{WZ}}, where the vector  superfield reduces to the form \DUrole{raw-tex}{$$V(z)=(0,0,0,\ A_{\mu},\ \lambda,\ D). $$}
The gauge is very convienient to write out supersymmetric Lagrangians. One interesting property of the Wess-Zumino
gauge is the ease with which powers of \DUrole{raw-tex}{$V(z)$} can be calculated \DUrole{raw-tex}{$(\ref{eq28})$}.

\begin{flalign}\label{eq28}
V(z)=\displaystyle \theta\sigma^{\mu}\overline{\theta}A_{\mu}(x)+\theta\theta\overline{\theta}\overline{\lambda}(x)+\overline{\theta}\overline{\theta}\theta\lambda(x)+\frac{1}{2}\theta\theta\overline{\theta}\overline{\theta}D(x) \nonumber \\
V^2(z) = \frac{1}{2}\theta\theta\overline{\theta}\overline{\theta}A^{\mu}A_{\mu},\nonumber \\
V^{n}(z)=0\forall n\geq 3
 \end{flalign}

The last point concerning the vector superfields is the expression of the field  strength which remains invariant under
supergauge transformation. Someone can construct left and right chiral field-strength superfields

\begin{flalign}\label{eq29}
 W_{A}=-\displaystyle \frac{1}{4}\bar{D}^{B}\bar{D}_{B}D_{A}V(x,\theta,\bar{\theta}) = -\frac{1}{4}\bar{D}\bar{D}D_{A}V, \nonumber \\
 \overline{W}_{A}=-\frac{1}{4}DD\overline{D}_{A}V
 \end{flalign}

\DUrole{raw-tex}{\noindent}
which, in accordance with \DUrole{raw-tex}{$(\ref{eq18})$},  obeys the following equations \DUrole{raw-tex}{$(\ref{eq30})$} in

\begin{flalign}\label{eq30}
\overline{D}_{A}W_{A}=0,\ \ D_{A}\overline{W}_{A}=0 \nonumber \\
 V^{'} = V+\Phi + \Phi^{\dagger}, \ \ W_A^{'} = \frac{1}{4}\bar{D}\bar{D}D_AV^{'}=
 \frac{1}{4}(\bar{D}(\{\bar{D},D_A\}-D_A\bar{D})(V+\Phi + \Phi^{\dagger}) = \nonumber \\
 \frac{1}{4}\bar{D}(\bar{D}D_AV+\bar{D}\Phi + D_A\Phi^{\dagger}) =  \frac{1}{4}\bar{D}\bar{D}D_AV = W_A
 \end{flalign}

\DUrole{raw-tex}{\noindent}
It is convenient to use \DUrole{raw-tex}{$y^{\mu}=x^{\mu}-i\theta\sigma^{\mu}\bar{\theta}$} or \DUrole{raw-tex}{$\overline{y}^{\mu}=x^{\mu}+i\theta\sigma^{\mu}\overline{\theta}$}
shifts to write out the vector field and the field strength  in the simplest way

\begin{flalign}\label{eq31}
 V(y,\displaystyle \ \theta,\overline{\theta})=\theta\sigma^{\mu}\overline{\theta}A_{\mu}(y)+\theta\theta\overline{\theta}\overline{\lambda}(y)+\overline{\theta}\overline{\theta}\theta\lambda(y)+\frac{1}{2}\theta\theta\overline{\theta}\overline{\theta}\{D(y)+i\partial_{\mu}^{(y)}A^{\mu}(y)\}\nonumber \\
 V(\displaystyle \overline{y},\ \theta,\overline{\theta})=\theta\sigma^{\mu}\overline{\theta}A_{\mu}(\overline{y})+\theta\theta\overline{\theta}\overline{\lambda}(\overline{y})+\overline{\theta}\overline{\theta}\theta\lambda(\overline{y})+\frac{1}{2}\theta\theta\overline{\theta}\overline{\theta}\{D(\overline{y})-i\partial_{\mu}^{(\overline{y})}A^{\mu}(\overline{y})\}\nonumber \\
 \overline{D}(y) = -\overline{\partial_A},\ \ \  D(y)=\partial_A - 2i\sigma^{\mu}_{AB}\overline{\theta}^B\partial_{\mu}, \ \ \ D(\overline{y}) = \partial_A\nonumber \\
 W_A = -\frac{1}{4}\bar{D}(y)\bar{D}(y)D_{A}(y)V(y,\theta,\bar{\theta})=
 \lambda_{A}(y) +D(y)\theta_{A}-(\sigma^{\mu\nu}\theta)_{A}F_{\mu\nu}(y)+i\theta\theta\sigma_{A\dot{B}}^{\mu}\partial_{\mu}^{(y)}\overline{\lambda}^{B}(y), \nonumber \\
 \overline{W}_A = \overline{\lambda}_{\dot{A}}(\overline{y})+D(\overline{y})\overline{\theta}_{\dot{A}}-(\overline{\sigma}^{\mu\nu}\overline{\theta})_{A}F_{\mu\nu}(\overline{y})-i\overline{\theta}\overline{\theta}\{\partial_{\mu}^{(\overline{y})}\lambda(\overline{y})\sigma^{\mu}\}_{A}
 \end{flalign}

\DUrole{raw-tex}{\noindent}
where  \DUrole{raw-tex}{$$F_{\mu\nu} = \partial_{\mu}A_{\nu} - \partial_{\nu}A_{\mu}$$}
The kinetic term \DUrole{raw-tex}{$1/4 |W_AW^A + \overline{W}_A\overline{W}^A|_{\theta\theta,\bar{\theta}\bar{\theta}}$} in the SUSY
Lagrangian takes the familar form \DUrole{raw-tex}{$(\ref{eq32})$}

\begin{flalign}\label{eq32}
 W^{A}W_{A}\ =\ \lambda(y)\lambda(y)+2\theta\{D(y)\lambda(y)+\sigma^{\mu\nu}\lambda(y)F_{\mu\nu}(y)\}+ \nonumber \\
 \theta\theta\{D^{2}(y)+2i\lambda(y)\sigma^{\mu}\partial_{\mu}^{(y)}\overline{\lambda}(y)-\frac{1}{2}F_{\mu\nu}(y)F^{\mu\nu}(y) -  \nonumber \\
 \displaystyle \frac{i}{2}\tilde{F}_{\mu\nu}(y)F^{\mu\nu}(y), \nonumber \\
 \overline{W}_{A}\overline{W}^{A}\ =\ \overline{\lambda}(\overline{y})\overline{\lambda}(\overline{y})+\{2D(\overline{y})\overline{\lambda}(\overline{y})
 +2\overline{\lambda}(\overline{y})\overline{\sigma}^{\mu\nu}F_{\mu\nu}(\overline{y})\}\overline{\theta} + \nonumber \\
 \overline{\theta}\overline{\theta}\{D^{2}(\overline{y})-2i\partial_{\mu}^{(\overline{y})}\lambda(\overline{y})\sigma^{\mu}\overline{\lambda}(y)-\frac{1}{2}F_{\mu\nu}(\overline{y})F^{\mu\nu}(\overline{y}) + \nonumber \\
 \displaystyle \frac{i}{2}\tilde{F}_{\mu\nu}(\overline{y})F^{\mu\nu}(\overline{y}),\nonumber \\
 \displaystyle \frac{1}{4}[W^{A}W_{A}+\overline{W}_{A}\overline{W}^{A}]_{\theta\theta,\overline\theta\overline\theta}=\frac{1}{2}D^{2}(x)-\frac{1}{4}F_{\mu\nu}(x)F^{\mu\nu}(x)+i\lambda(x)\sigma^{\mu}[\partial_{\mu}]\overline{\lambda}(x)
 \end{flalign}

\DUrole{raw-tex}{\noindent}
In the last line of \DUrole{raw-tex}{$(\ref{eq32})$}, the back substitution \DUrole{raw-tex}{$y\rightarrow x$} was done in \DUrole{raw-tex}{$F-$} term because
the 4-dimensional intergral
is invariant under all SUSY transformations \DUrole{raw-tex}{$\int d^4x W^{A}W_{A}|_{\theta\theta}  \equiv \int d^4y W^{A}W_{A}|_{\theta\theta}$}


\subsubsection{2.2.3~~~R-parity of the chiral and vector superfields%
  \label{r-parity-of-the-chiral-and-vector-superfields}%
}

There is \DUrole{raw-tex}{$R-$} symmetry as a global \DUrole{raw-tex}{$U(1)$} transformation in SUSY which leaves Super-Poincare Lie Algebra \DUrole{raw-tex}{$(\ref{eq11}),(\ref{eq12}),(\ref{eq13}),(\ref{eq14})$}
unchanged.

 \begin{flalign}\label{eq33}
 Q_{A}\rightarrow e^{i\varphi R}Q_{A}e^{-i\varphi R}=e^{-i\varphi}Q_{A}, \nonumber\\
\overline{Q}_{A}\rightarrow e^{i\varphi R}\overline{Q}_{A}e^{-i\varphi R}=e^{i\varphi}\overline{Q}_{A}, \nonumber\\
 \theta\rightarrow e^{i\varphi}\theta,\ \ \ \overline{\theta} \rightarrow e^{-i\varphi}\overline{\theta}, \nonumber \\
[Q_A,R]=Q_A , \nonumber \\
[\overline{Q}_A,R]=-\overline{Q}_A
 \end{flalign}

\DUrole{raw-tex}{\noindent}
This \DUrole{raw-tex}{$(\ref{eq33})$} leads that  new quantum numbers, \DUrole{raw-tex}{$R-$} charges, can be assigned to
\DUrole{raw-tex}{$\theta,\bar{\theta},Q,\bar{Q},\Phi,\Phi^{\dagger}$} ,
which are \DUrole{raw-tex}{$1,-1,-1,1,R_{\Phi},-R_{\Phi}$} respectively, where  \DUrole{raw-tex}{$R_{\Phi}$} are the \DUrole{raw-tex}{$R-$} charges for the left and
rifht chiral superfields  derived from their \DUrole{raw-tex}{$R-$} symmetry properties

\begin{flalign}\label{eq34}
\Phi^{\prime}(x,e^{i\varphi}\theta,e^{-i\varphi}\overline{\theta})\rightarrow e^{i\varphi R_{\Phi}}\Phi(x,\theta,\overline{\theta}), \nonumber\\
\Phi^{\prime\dagger}(x,e^{i\varphi}\theta,e^{-i\varphi}\overline{\theta})\rightarrow e^{-i\varphi R_{\Phi}}\Phi^{\dagger}(x,\theta,\overline{\theta}), \nonumber\\
 \end{flalign}

\DUrole{raw-tex}{\noindent}
The vector field has zero \DUrole{raw-tex}{$R-$} charge because of its reality \DUrole{raw-tex}{$V(x)=V^{\dagger}(x)$}.
Someone also can write out \DUrole{raw-tex}{$R-$} charges of the chiral and vector components in \emph{Wees-Zummino gauge} using \DUrole{raw-tex}{$(\ref{eq34})$}
and the previous statement about \DUrole{raw-tex}{$V(x)$}

\begin{flalign}\label{eq35}
R(\phi)=R_{\Phi}, \nonumber \\
R(\xi)=-R(\overline{\xi})=R_{\Phi}-1, \nonumber \\
R(F)=R_{\Phi}-2, \nonumber \\
R(A_{\mu})=0, \nonumber \\
R(\lambda)=-R(\overline{\lambda})=1, \nonumber \\
R(D)=0
 \end{flalign}

\DUrole{raw-tex}{\noindent}

The supersymmetric Lagrangian with gauge interaction has a general form \DUrole{raw-tex}{$\ref{eq36}$}

\begin{flalign}\label{eq36}
 \mathcal{L} = \frac{1}{4}(\int d^2\theta W_AW^A + \int d^2\overline{\theta}\overline{W_A}\overline{W^A})  + \nonumber \\
 \int d^2\theta d^2\overline{\theta} \Phi^{\dagger}e^{V}\Phi + \int d^2\theta W(\Phi) + \int d^2\theta W^{*}(\Phi^{\dagger})
 \end{flalign}

\DUrole{raw-tex}{\noindent}
The kinetic \DUrole{raw-tex}{$W_AW^A|_{\theta\theta}$} and gauge interaction \DUrole{raw-tex}{$\Phi^{\dagger}e^V\Phi|_{\theta\theta\overline{\theta}\overline{\theta}}$}
terms are \DUrole{raw-tex}{$R-$} invariant, ie they have total zero \DUrole{raw-tex}{$R-$} charge.  But the superpotentional \DUrole{raw-tex}{$W(\Phi)$}, some analytic function of  having 3 \textsuperscript{rd} power of \DUrole{raw-tex}{$\Phi(x)$} at most.
and describing the self-interactions of superfields, can preserve or not preserve \DUrole{raw-tex}{$R-$} invariance. The \DUrole{raw-tex}{$R-$} nonsymmetric
terms can be presented in the part of the Lagrangian correspoding to  Yukawa interaction  as well in soft supersymmetry  breaking
terms. The Yukawa  \DUrole{raw-tex}{$R_p-$} violating  terms are forbidden in SUSY generalization of the Standard Model because of their non-Lorentz
invariance. In the time, the soft supersimmetry breaking part contains Lorentz and gauge invariant mass terms of gaugino,
Majorana spinor \DUrole{raw-tex}{$\lambda,\overline{\lambda}$} component of the vector superfield \DUrole{raw-tex}{$V(x)$}
\DUrole{raw-tex}{$$ M\cdot(\overline{\lambda}\overline{\lambda} + \lambda\lambda), $$} which has \DUrole{raw-tex}{$R\neq 0.$}
Thus \DUrole{raw-tex}{$U(1)_R$} invariance  has to be too restrictive and \DUrole{raw-tex}{$U(1)_R$} transformaton has to be abandoned as symmetry but
its discrete subgroup \DUrole{raw-tex}{$Z_2$} which has only two possible group elements \DUrole{raw-tex}{$\varphi=\pm \pi$}, conserves
soft supersymmetric breaking terms \DUrole{raw-tex}{$$ Z_2: \,\,\, M\lambda^{\prime}\lambda^{\prime}=Me^{i2\pi}\lamda\lambda=M\lambda\lambda.$$}
The value of \DUrole{raw-tex}{$e^{i\pi R_{\Phi}}=(-1)^{R_{\Phi}}$} is called a matter parity \DUrole{raw-tex}{$M_p$} of the superfield, while the corresponded
value for the component field is called  \DUrole{raw-tex}{$R_p-$} parity of the that field. In accordance to \DUrole{raw-tex}{$(\ref{eq35})$} and
\DUrole{raw-tex}{$Z_2$} symmetry,
the values of \DUrole{raw-tex}{$R_{\Phi}$} for field components can be \DUrole{raw-tex}{$\pm 1`$ or `$0$}. The two options define two categories of the chiral superfields:
matterlike or quantalike. In both cases, component fields corresponded to \emph{Standard Model} (\emph{SM})
representations have positive values of
\DUrole{raw-tex}{$R_p$} or equally  the component field has \DUrole{raw-tex}{$R_{\Phi}=0$}. Such supermultiplets, which contain spin 1/2 \emph{SM}
particles, as quarks or leptons, but scalar superpartners, are matterlike.  Oppositely, the quantalike multiples consist of
bosonic \emph{SM} fields and fermionice sparticles. This property of \DUrole{raw-tex}{$R_{\Phi}$} can be ensured by the relation \DUrole{raw-tex}{$$R_{\Phi}=3(B-L),$$}
where \DUrole{raw-tex}{$B$} is a baryon number and \DUrole{raw-tex}{$L-$} lepton number associated with the superfield. Then \DUrole{raw-tex}{$M_p$} parity
is \DUrole{raw-tex}{$(-1)^{3(B-L)}$} while
\DUrole{raw-tex}{$R_p-$} parity is expressed as \DUrole{raw-tex}{$(-1)^{3(B-L)+2S},$} with \DUrole{raw-tex}{$S$} being the spin.

\begin{flalign}\label{eq366}
M_p=(-1)^{3(B-L)},\,\,\,\, R_p=(-1)^{3(B-L)+2S}
\end{flalign}

\DUrole{raw-tex}{\noindent}
\DUrole{raw-tex}{$R_p-$} parity of a state containing several particles and spaiticles is the product of the individual parities.Thus
for the particles and sparticles under discussion,  \DUrole{raw-tex}{$R_{p}$} conservation in any interaction vertex
means that sparticle production processes must produce them in even numbers (usually a pair) and
every sparticle other than the \emph{Lightest Supersymmetric Particle} (LSP) will eventually decay into particles plus
an odd number of LSPs (usually one). I will consider  \emph{Minimal Supersymmetric Standard Model}  (MSSM) as   a such supersymmetry theory
with exact \DUrole{raw-tex}{$R-$} parity conservation.

\setlength{\DUtablewidth}{\linewidth}
\begin{longtable}[c]{|p{0.185\DUtablewidth}|p{0.269\DUtablewidth}|p{0.058\DUtablewidth}|p{0.111\DUtablewidth}|p{0.164\DUtablewidth}|p{0.174\DUtablewidth}|}
\caption{\DUrole{raw-tex}{$R_p$} for particles and sparticles}\\
\hline
\textbf{%
Supermultiplet
} & \textbf{%
Name
} & \textbf{%
Spin
} & \textbf{%
\DUrole{raw-tex}{$R_p$}
} & \textbf{%
\DUrole{raw-tex}{$R_{\Phi}$}
} & \textbf{%
Superfild type
} \\
\hline
\endfirsthead
\caption[]{\DUrole{raw-tex}{$R_p$} for particles and sparticles (... continued)}\\
\hline
\textbf{%
Supermultiplet
} & \textbf{%
Name
} & \textbf{%
Spin
} & \textbf{%
\DUrole{raw-tex}{$R_p$}
} & \textbf{%
\DUrole{raw-tex}{$R_{\Phi}$}
} & \textbf{%
Superfild type
} \\
\hline
\endhead
\multicolumn{6}{c}{\hfill ... continued on next page} \\
\endfoot
\endlastfoot

Particle

Sparticle
 & 
Quark,q

Squark, \DUrole{raw-tex}{$\tilde{q}$}
 & 
1/2

0
 & 
+1

-1
 & 
1

1
 & 
Chiral,

matterlike
 \\
\hline

Particle

Sparticle
 & 
Lepton, l

Slepton, \DUrole{raw-tex}{$\tilde{l}$}
 & 
1/2

0
 & 
+1

-1
 & 
1

1
 & 
Chiral,

matterlike
 \\
\hline

Particle

Sparticle
 & 
Higgs,H

Higgsino, \DUrole{raw-tex}{$\tilde{H}$}
 & 
0

1/2
 & 
+1

-1
 & 
0

0
 & 
Chiral,

quantalike
 \\
\hline

Particle

Sparticle
 & 
Gauge boson,g

Gaugino, \DUrole{raw-tex}{$\tilde{g}$}
 & 
1

1/2
 & 
+1

-1
 & 
0

0
 & 
Vector
 \\
\hline
\end{longtable}


\subsection{2.3~~~Construction of SUSY Lagrangians%
  \label{construction-of-susy-lagrangians}%
}

The general SUSY  Lagrangian \DUrole{raw-tex}{$(\ref{eq37})$} consists of the kinetic term, gauge interaction and   superpotential as it follows
from \DUrole{raw-tex}{$(\ref{eq36})$}.

\begin{flalign}\label{eq37}
 \mathcal{L} = \frac{1}{4} (W_AW^A|_{\theta\theta} + \overline{W_A}\overline{W^A}|_{\overline{\theta}\overline{\theta}}) + \nonumber \\
  \Phi^{\dagger}e^{V}\Phi|_{\theta\theta\overline{\theta}\overline{\theta}} + W(\Phi)|_{\theta\theta}
 \end{flalign}

\DUrole{raw-tex}{\noindent}
The superpotential \DUrole{raw-tex}{$W(\Phi)$} is a polinomial  of chiral superfields which Taylor expansion looks like

\begin{flalign}\label{eq38}
 W(\Phi_{i})\ =\ W(\phi_{i}+\sqrt{2}\theta\xi_{i}+\theta\theta F) \nonumber \\
 = \displaystyle W(\phi_{i})+\frac{\partial W}{\partial \phi_{i}}\sqrt{2}\theta\xi_{i}+\theta\theta(  \nonumber \\
 \frac{\partial W}{\partial \phi_{i}}F_{i}-\frac{1}{2}\frac{\partial^{2}W}{\partial \phi_{i}\partial \phi_{j}}\xi_{i}\xi_{j}).
\end{flalign}

\DUrole{raw-tex}{\noindent}
In particular, the SUSY generalization of QED, with two chiral supermultiplets introduced to have left- and right-handed
fermions  and with  the superpotential  of the second order \DUrole{raw-tex}{$W(\Phi_{-},\Phi_{+})=m\Phi_{-}\Phi_{+}$} is

 \begin{flalign}\label{eq39}
 \mathcal{L}_{SUSYQED}\ =\ \frac{1}{4}(W^{\alpha}W_{\alpha}|_{\theta\theta}+W^{\alpha}W_{\alpha}|_{\overline{\theta}\overline{\theta}})\nonumber \\
+ \displaystyle (\Phi_{+}^{+}e^{gV}\Phi_{+}+\Phi_{-}^{+}e^{-gV}\Phi_{-})|_{\theta\theta\overline{\theta}\overline{\theta}} \nonumber \\
 +\  m(\Phi_{+}\Phi_{-}|_{\theta\theta} + \Phi_{+}^{+}\Phi_{-}^{+}|_{\overline{\theta}\overline{\theta}})
 \end{flalign}

\DUrole{raw-tex}{\noindent}
Someone can get the expression of SUSY Lagrangian  explicitly in terms of component fields.
The  simplest SUSY  model proposed by  Wess and Zumino  \DUrole{raw-tex}{\cite{Wess:1974jb},\cite{Wess:1974tw}}
has the  Lagrangian of only one
left chiral superfield   without  gauge  interaction.
The common expression of \DUrole{raw-tex}{$\mathcal{L}_{SUSY}$}
\DUrole{raw-tex}{$$ \mathcal{L}_{SUSY}\ =   \Phi^{\dagger}_{i}\Phi_{j}|_{\overline{\theta\theta}\theta\theta} +W(\Phi_{i}\Phi_{j})|_{\theta\theta} $$}

depends on the  kinetic and interaction terms given by \DUrole{raw-tex}{$(\ref{eq40})$}.and \DUrole{raw-tex}{$(\ref{eq41})$}.

\begin{flalign}\label{eq40}
\Phi_{i}^{\dagger}\Phi_{j}\ =\ \phi_{\mathrm{i}}^{\star}\phi_{j}+\sqrt{2}\theta\xi_{j}\phi_{i}^{\star}+\sqrt{2}\overline{\theta}\overline{\xi}_{i}\phi_{j}+\theta\theta\phi_{i}^{\star}F_{j}+\overline{\theta}\overline{\theta}F_{i}^{\star}\phi_{j}+2\overline{\theta}\overline{\xi}_{i}\theta\xi_{j} \nonumber \\
 +\sqrt{2}\theta\theta\overline{\theta}_{\dot{A}}(i\overline{\sigma}^{\mu\dot{A}B}\xi_{jB}[\partial_{\mu}]\phi_i^{\star}+\overline{\xi_i^{\dot{A}}}F_j)\nonumber \\
 -2i\theta\sigma^{\mu}\overline{\theta}\phi_{i}^{\star}[\partial_{\mu}]\phi_{j} \nonumber \\
 +\sqrt{2}\overline{\theta}\overline{\theta}\theta^{A}(i\sigma_{A\dot{B}}^{\mu}\overline{\xi}_{i}^{B}[\partial_{\mu}]\phi_{j}+\xi_{jA}F_{i}^{\star}) \nonumber \\
  + \theta\theta\overline{\theta}\overline{\theta}(F_{i}^{\star}F_{j}+\frac{1}{2}\partial_{\mu}\phi_{i}^{\star}[\partial^{\mu}]\phi_{j}-\frac{1}{2}\phi_{i}^{\star}[\partial_{\mu}]\partial^{\mu}\phi_{j}+i\xi_{j}\sigma^{\mu}[\partial_{\mu}]\overline{\xi}_{i}), \nonumber  \\
\Phi_{i}^{\dagger}\Phi_{j}|_{\overline{\theta\theta}\theta\theta} = F_{i}^{\star}F_{j}+\frac{1}{2}\partial_{\mu}\phi_{i}^{\star}[\partial^{\mu}]\phi_{j}-\frac{1}{2}\phi_{i}^{\star}[\partial_{\mu}]\partial^{\mu}\phi_{j}+i\xi_{j}\sigma^{\mu}[\partial_{\mu}]\overline{\xi}_{i},
\end{flalign}

\begin{flalign}\label{eq41}
 W(\Phi) = \lambda_{i}\Phi_{i}+\frac{1}{2}m_{ij}\Phi_{i}\Phi_{j}+\frac{1}{3}g_{ijk}\Phi_{i}\Phi_{j}\Phi_{k}, \nonumber \\
 W(\Phi)|_{\theta\theta} = \lambda_{i}F_{i} + \frac{1}{2}m_{ij}(\phi_{i}F_{j}+\phi_{j}F_{i}-\xi_{i}\xi_{j}) \nonumber \\
 +  \frac{1}{3} g_{ijk}  (F_{i}\phi_{j}\phi_{k}+F_{j}\phi_{k}\phi_{i}+F_{k}\phi_{i}\phi_{j}-\xi_{i}\xi_{j}\phi_{k}-\xi_{j}\xi_{k}\phi_{i}-\xi_{k}\xi_{i}\phi_{j})
\end{flalign}

\DUrole{raw-tex}{\noindent}
The superpotential  is a polinomial of the chiral superfield \DUrole{raw-tex}{$\Phi_i(z)$} of third order \DUrole{raw-tex}{$(\ref{eq41})$} at most that make SUSY to be  regularized.
The auxiliary fields \DUrole{raw-tex}{$F_{i}$} can ruled be out from \DUrole{raw-tex}{$(\ref{eq40})$} by the equtation of motion \DUrole{raw-tex}{$(\ref{eq42})$}

\begin{flalign}\label{eq42}
\frac{\partial \mathcal{L}}{\partial F_{k}^{*}} = F_{k}+\lambda_{k}^{*}+m_{ik}^{*}\phi_{i}^{*}+y_{ijk}^{*}\phi_{i}^{*}\phi_{j}^{*}=0, \nonumber \\
 \frac{\partial \mathcal{L}}{\partial F_{k}} = F_{k}^{*}+\lambda_{k}+m_{ik}\phi_{i}+y_{ijk}\phi_{i}\phi_{j}=0
\end{flalign}

\DUrole{raw-tex}{\noindent}
The superpotential  \DUrole{raw-tex}{$W(\Phi)$} in Wess-Zumino's model,  \DUrole{raw-tex}{$\mathcal{L}_{SUSY}$}, is supposed
to have  the form \DUrole{raw-tex}{$$W(\Phi)=\frac{1}{2}m\Phi\Phi+\frac{g}{3}\Phi\Phi\Phi+h.c.$$} .
Using Taylor expansion \DUrole{raw-tex}{$(\ref{eq38})$}  of the superpotential \DUrole{raw-tex}{$W(\Phi)$}, the equation of
motion \DUrole{raw-tex}{$(\ref{eq42})$} is transformed to the simplest form as shown in  \DUrole{raw-tex}{$(\ref{eq43})$}

\begin{flalign}\label{eq43}
 W(\Phi_{i})\ =\ \mathcal{W}(\phi_{i}+\sqrt{2}\theta\xi_{i}+\theta\theta F_i)=\nonumber \\
 \displaystyle W(\phi_{i})+\frac{\partial \mathcal{W}}{\partial \phi_{i}}\sqrt{2}\theta\xi_{i}+\theta\theta(\frac{\partial \mathcal{W}}{\partial \phi_{i}}F_{i}-\frac{1}{2}\frac{\partial^{2}\mathcal{W}}{\partial \phi_{i}\partial \phi_{j}}\xi_{i}\xi_{j}), \nonumber \\
 \mathcal{L}_F|_{\theta\theta} = F_{i}^{\star}F_{i} + (\frac{\partial W}{\partial \phi_{i}}F_{i}+\frac{\partial W}{\partial \phi_{i}^{\star}}F_{i}^{\star}) -\frac{1}{2}\frac{\partial^{2}\mathcal{W}}{\partial \phi_{i}\partial \phi_{j}}\xi_{i}\xi_{j}, \nonumber \\
 \frac{\partial \mathcal{L}_F|_{\theta\theta}}{\partial F_i} = 0\Rightarrow F_i^{\star} = -\frac{\partial \mathcal{W}}{\partial \phi_{i}}, \nonumber \\
 \frac{\partial \mathcal{L}_F|_{\theta\theta}}{\partial F_i} = 0\Rightarrow F_i = -\frac{\partial \mathcal{W}}{\partial \phi_{i}^{\star}}, \nonumber \\
  F_i = -m\phi^{\star}_i - g\phi_i^{\star}\phi_i^{\star},\,\, F_i^{\star} = -m\phi_i - g\phi_i\phi_i.
\end{flalign}

\DUrole{raw-tex}{\noindent}
Making the substitution of \DUrole{raw-tex}{$F_i$}  derived  in  \DUrole{raw-tex}{$(\ref{eq43})$}, someone can infer that
SUSY Lagrangian \DUrole{raw-tex}{$\mathcal{L}_{SUSY}$} describes   a Dirac massive fermion
field \DUrole{raw-tex}{$\xi(x)$}, a complex scalar field \DUrole{raw-tex}{$\phi(x)=1/\sqrt{2}(h(x)+iH(x))$} with scalar and pseudoscalar \DUrole{raw-tex}{$h,H$} as
mass eigenbasis of the Higgs sector determined by \DUrole{raw-tex}{$\phi(x)$}. The full SUSY Lagragngian of the model
\DUrole{raw-tex}{\cite{{Wess:1974jb},\cite{Wess:1974t}}  is  given by \DUrole{raw-tex}{$(\ref{eq44})$} in terms of the physical fields.
The Higgs sector is defined by the scalar potential \DUrole{raw-tex}{$V(h,H)$}
\DUrole{raw-tex}{$(\ref{eq45})$} having the mass  and self-interaction terms of the \DUrole{raw-tex}{$h$} and \DUrole{raw-tex}{$H$} fields.
The Yukawa interaction between  \DUrole{raw-tex}{$\xi$} fermion  and  \DUrole{raw-tex}{$\phi$} scalar fields
is given by  Lagrangian \DUrole{raw-tex}{$\mathcal{L}_I$} in  \DUrole{raw-tex}{$(\ref{eq44})$}. The other parts of SUSY Lagrangian \DUrole{raw-tex}{$\mathcal{L}_K$} and
\DUrole{raw-tex}{$\mathcal{L}_M$} are the kinetic and mass terms of the Lagrangian.

\begin{flalign}\label{eq44}
\mathcal{L}_K=\frac{1}{2} \partial _{\mu }[h]{}{}^2+\frac{1}{2} \partial _{\mu }[H]{}{}^2- \frac{1}{2} h \partial _{\mu }\left[\partial _{\mu }[h]\right]-\frac{1}{2} H \partial _{\mu }\left[\partial _{\mu }[H]\right]\nonumber \\
-\frac{1}{2} i \partial _{\mu }\left[\overset{-}{\xi }_{\alpha }\right].\xi _{\beta} \gamma {}^{\mu }.P^-_{\alpha ,\beta}+\frac{1}{2} i \overset{-}{\xi }_{\alpha }.\partial _{\mu }\left[\xi _{\beta }\right] \gamma {}^{\mu }.P^-_{\alpha ,\beta }-\nonumber \\
 \frac{1}{2} i \partial _{\mu }\left[\overset{-}{\xi }_{\beta }\right].\xi _{\alpha } \gamma {}^{\mu }.P^+_{\beta ,\alpha }+\frac{1}{2} i \overset{-}{\xi }_{\beta }.\partial _{\mu }\left[\xi _{\alpha }\right] \gamma {}^{\mu }.P^+_{\beta ,\alpha },\nonumber  \\
 \mathcal{L}_M=-\frac{1}{2} h{}^2 m{}^2-\frac{H{}^2 m{}^2}{2}-\frac{1}{4} m \overset{-}{\xi }_{\alpha }{}^C.\xi _{\beta } P^-_{\alpha ,\beta }-\nonumber \\
 \frac{1}{4} m \overset{-}{\xi }_{\beta }{}^C.\xi _{\alpha } P^-_{\beta ,\alpha }-\frac{1}{4} m \overset{-}{\xi }_{\alpha }.\xi _{\beta }{}^C P^+_{\alpha ,\beta }-\frac{1}{4} m \overset{-}{\xi }_{\beta }.\xi _{\alpha }{}^C P^++_{\beta ,\alpha },\nonumber \\
 \mathcal{L}_I=-\frac{1}{4} g{}^2 h{}^4-\frac{1}{2} g{}^2 h{}^2 H{}^2-\frac{g{}^2 H{}^4}{4}-\frac{g h{}^3 m}{\sqrt{2}}-\frac{g h H{}^2 m}{\sqrt{2}}-\frac{g h \overset{-}{\xi }_{\alpha }{}^C.\xi _{\beta } P^-_{\alpha ,\beta }}{2 \sqrt{2}}\nonumber \\
-\frac{i g H \overset{-}{\xi }_{\alpha }{}^C.\xi _{\beta } P^-_{\alpha ,\beta }}{2 \sqrt{2}}-
\frac{g h \overset{-}{\xi }_{\beta }{}^C.\xi _{\alpha } P^-_{\beta ,\alpha }}{2 \sqrt{2}}-\frac{i g H \overset{-}{\xi }_{\beta }{}^C.\xi _{\alpha } P^-_{\beta ,\alpha }}{2 \sqrt{2}}\nonumber \\
-\frac{g h \overset{-}{\xi }_{\alpha }.\xi _{\beta }{}^C P^+_{\alpha ,\beta }}{2 \sqrt{2}}+\frac{i g H \overset{-}{\xi }_{\alpha }.\xi _{\beta }{}^C P^+_{\alpha ,\beta }}{2 \sqrt{2}}-\frac{g h \overset{-}{\xi }_{\beta }.\xi _{\alpha }{}^C P^+_{\beta ,\alpha }}{2 \sqrt{2}}\nonumber \\
+\frac{i g H \overset{-}{\xi }_{\beta }.\xi _{\alpha }{}^C P^+_{\beta ,\alpha }}{2 \sqrt{2}}
\end{flalign}

\DUrole{raw-tex}{\noindent}
Here \DUrole{raw-tex}{$\xi^C$} is the charge-conjugated field, and \DUrole{raw-tex}{$P^{\pm}$} denotes the left- and right-handed projective operators
\DUrole{raw-tex}{$$P^{\pm}=\frac{1_{4x4}\pm \gamma^5}{2}$$}

\begin{flalign}\label{eq45}
V(h,H)= \frac{g^2 h^4}{4}+\frac{1}{2} g^2 h^2 H^2+\frac{g^2
H^4}{4}+\frac{g h^3 m}{\sqrt{2}}+\frac{g h H^2
m}{\sqrt{2}}+\frac{h^2 m^2}{2}+\frac{H^2 m^2}{2}
\end{flalign}

\DUrole{raw-tex}{\noindent}
The vacum states  of the quantum scalar field  system  are ones which satisfies
the minimum  of the scalar potential \DUrole{raw-tex}{$V(h,H)$} . The  solutions for such vacum expectation values  (v.e.v.s) of \DUrole{raw-tex}{$h,H$} are
shown in the equation \DUrole{raw-tex}{$(\ref{eq46})$}  and Fig. \DUrole{raw-tex}{$\ref{fig6}$} shows \DUrole{raw-tex}{$V(h,H)$} as a function  of  the scalar boson amplitude
\DUrole{raw-tex}{$|\phi|$} \DUrole{raw-tex}{$(\ref{eq47})$} .

 \begin{flalign}\label{eq46}
 h\to 0,H\to 0, \nonumber \\
 h\to -\frac{\sqrt{2} m}{g},H\to 0, \nonumber \\
 h\to 0,H\to \frac{\sqrt{2} m (\pm i)}{g}, \nonumber \\
\end{flalign}

\begin{flalign}\label{eq47}
\phi(x)=|\phi(x)|e^{i\cdot\alpha}, \,\, h(x)=Re(\phi(x)),\,\, H(x)=i\cdot Im(\phi(x))
\end{flalign}
\begin{figure}
\noindent\makebox[\textwidth][c]{\includegraphics[width=0.800\linewidth]{/mnt/WorkingPlace/Thesis/thesis/theory/susy/plots/WeesZuminoScalarPotential.png}}
\caption{The scalar potential \DUrole{raw-tex}{$V(h,H)$} as a function of the amplitude \DUrole{raw-tex}{$|\phi|$} and the phase \DUrole{raw-tex}{$\alpha$} of the Higgs field \DUrole{raw-tex}{\label{fig6}}}
\end{figure}

\DUrole{raw-tex}{\noindent}
Such non zero v.e.v.s of \DUrole{raw-tex}{$h,H$} when \DUrole{raw-tex}{$V$} has the minimum  is non-SUSY ground state which is  appeared
after spontaneous SUSY breaking.
This non-SUSY ground state  finally leads to the gauge symmetry  breaking ( \DUrole{raw-tex}{$U(1)$} in the case of the model
\DUrole{raw-tex}{\cite{Wess:1974jb}} , \DUrole{raw-tex}{\cite{Wess:1974tw}} ) breaking via the Higgs mechanism. On the other hand the SUSY breaking is resulting  from  auxiliary fields  \DUrole{raw-tex}{$F_i$} or
\DUrole{raw-tex}{$D_i$}  which occur to have non-zero v.e.v.s. as solutions of  the  equation of motion. There is an example of such \DUrole{raw-tex}{$F-$}
mechanish \DUrole{raw-tex}{\cite{O'Raifeartaigh:1975pr}}  shown in \DUrole{raw-tex}{$(\ref{eq48})$} .
In this case, the superpotential of the several chiral fields \DUrole{raw-tex}{$(\ref{eq48})$} doesn't provide
the   solutions consitent  with \DUrole{raw-tex}{$<0|F_i|0>=0$} for  some chiral supermutiplets and therefore SUSY is spontaneously
broken.

\begin{flalign}\label{eq48}
W(\Phi) = \lambda \Phi_3 + m\Phi_1\Phi_2 + g\Phi3\Phi_1^2 \Rightarrow \nonumber \\
F_1^{\star} = m\phi_2 + 2g\phi_1\phi_3,\,\,\,  F_2^{\star} = m\phi_1,\,\,\, F_3^{\star} = \lambda + g\phi_1^2
\end{flalign}

\DUrole{raw-tex}{\noindent}


\subsubsection{2.3.1~~~Soft-SUSY breaking%
  \label{soft-susy-breaking}%
}

The  described SUSY  model \DUrole{raw-tex}{\cite{Wess:1974jb},\cite{Wess:1974tw}} doesn't contain quadratic divergences destabilizing the scalar (higgs) sector.
Such divergences appear in the tadpoles and the selfenergies of the Higgs fields as it is illustrated by Fig. \DUrole{raw-tex}{$\ref{fig7},\ref{fig8}$}
\begin{figure}
\noindent\makebox[\textwidth][c]{\includegraphics[width=0.800\linewidth]{/mnt/WorkingPlace/Thesis/thesis/theory/susy/plots/WeesZuminoTadpole.PNG}}
\caption{One-loop contribution to the wave function  of  \DUrole{raw-tex}{$h(x)$} \DUrole{raw-tex}{\label{fig7}}}
\end{figure}
\begin{figure}
\noindent\makebox[\textwidth][c]{\includegraphics[width=0.500\linewidth]{/mnt/WorkingPlace/Thesis/thesis/theory/susy/plots/WeesZuminoSelfEnergy.PNG}}
\caption{One-loop contribution to the propagator of \DUrole{raw-tex}{$h(x)$} \DUrole{raw-tex}{\label{fig8}}}
\end{figure}

\DUrole{raw-tex}{\noindent}
The one-loop contributions can be estimated. A simple calculation gives  the equations \DUrole{raw-tex}{$(\ref{eq49})$} in the
dimensional regularization \DUrole{raw-tex}{\cite{'tHooft:1972fi},\cite{Bollini:1972ui}, \cite{Bollini:1972ui}}. The loop integrals introduced in
\DUrole{raw-tex}{$(\ref{eq49})$}  are reduced to the scalar  one-point \DUrole{raw-tex}{$A_0(m^2)$} and two-point \DUrole{raw-tex}{$B_{0(1)}(p^2,m_1^2,m_2^2)$} functions
using tensor decomposition technique explained in \DUrole{raw-tex}{\cite{Passarino:1978jh}}.
The functions are ultra-violet (UV) divergent and their definitions can be found  in the \DUrole{raw-tex}{\cite{'tHooft:1978xw}} work..

\begin{flalign}\label{eq49}
 Tadpole\sim <0|\mathcal{L}_I|h>=-\frac{9 g^2 \text{A}_0\left(M_h^2\right) m^2}{16 M_h^2 \pi ^2} -\frac{3 g^2\text{A}_0(M_H^2) m^2}{16 M_h^2 \pi^2} + \frac{3 g^2\text{A}_0(M_{\xi}^2) m M_{\xi}}{4 M_h^2 \pi^2},\nonumber \\
 SelfEnergy\sim <h|\mathcal{L}_I|h>= \frac{3 A_0(M_h^2) g^2}{16 \pi ^2}+\frac{A_0(M_H^2) g^2}{16\pi^2}-\frac{A0(M_{\xi}^2) g^2}{4 \pi ^2}+ \nonumber \\
 \frac{3 m^2  B_0(M_h^2,M_h^2,M_h^2) g^2}{16 \pi ^2}+\frac{m^2 B_0(M_h^2,M_H^2,M_H^2) g^2}{16 \pi^2} - \nonumber \\
-\frac{g^2 M_{\xi}^2 B0(M_h^2,M_{\xi}^2,M_{\xi}^2)}{4 \pi^2}  - p_{\mu}\frac{M_h^2 g^2 B_1(M_h^2,M_{\xi}^2,M_{\xi}^2)}{4 \pi^2}
\end{flalign}

\DUrole{raw-tex}{\noindent}
Someone can explicitly show that \DUrole{raw-tex}{$(\ref{eq49})$} contains only \DUrole{raw-tex}{$\log (\Lambda^2/M_{h(H,\xi)}^2)$} divergent terms
\DUrole{raw-tex}{$(\ref{eq50})$} , if  \texttt{cut-off} regularization \DUrole{raw-tex}{\cite{Kleinert:2001hn},\cite{Polchinski:1983gv}} is applied to \DUrole{raw-tex}{$(\ref{eq49})$} . .

\begin{flalign}\label{eq50}
 A_0(m^2) \rightarrow m^2(\Lambda^2/m^2 - \log (\Lambda^2/m^2)),\,\, B_0(p^2,m^2,m^2)\rightarrow \log (\Lambda^2/m^2),\nonumber \\
 Tadpole=\frac{3 g^2 m } { 16 M_h^2 \pi^2} (-4 m \Lambda^2 + 4 M_{\xi}\Lambda^2     ,\nonumber \\
 + 3 M_h^2 m \log (\Lambda^2/M_h^2)  + M_H^2 m \log (\Lambda^2/M_H^2) - 4 M_{\xi}^3 \log (\Lambda^2/M_{\xi}^2), \nonumber \\
 SelfEnergy= \frac{g^2}{16 \pi^2}[(3m^2 - 3 M_h^2)\log (\Lambda^2/M_h^2) + (m^2 - M_H^2) \log (\Lambda^2/M_H^2)]
\end{flalign}

\DUrole{raw-tex}{\noindent}
The quadratic divergency is canceled only if the fermion mass \DUrole{raw-tex}{$M_{\xi}$}  is equal to  the parameter \DUrole{raw-tex}{$m$}
which determines  the mass term of the superpotential \DUrole{raw-tex}{$W(\Phi_i)$}. The case,  when the masses
of the scalars and fermions  after SUSY and gauge  symmetry breaking  are the same, \DUrole{raw-tex}{$M_H=M_h=M_{\xi}=m$} ,
corresponds to the total cancelation of the divergency in the one-loop effects.
Also it turns out that it is  possible to  extend  the superpotential \DUrole{raw-tex}{$W(\Phi_i)$} by terms \DUrole{raw-tex}{$\sim \lambda\Phi_i + m^{\prime}\Phi^2_i+ g^{\prime}\Phi^3_i $}
in such way that SUSY symmetry  is broken but only logarithmic divergences \DUrole{raw-tex}{$(\ref{eq50})$} will be presented at the one-loop level.
This is  the \texttt{soft-SUSY} breaking. MSSM is the theory with  the \texttt{soft-SUSY} breaking mechanism.


\section{3~~~The Minimal Supersymmetric Model%
  \label{the-minimal-supersymmetric-model}%
}

The SUSY theories has the qual number of bosonic and fermionic degrees of freedom. Minimal version of SUSY generalization  of
SM (MSSM) doubles the number of particles, introducing a superpartner to each particle \DUrole{raw-tex}{\cite{MSSM1}}. Unfortunately, the  \DUrole{raw-tex}{$F(D)-$} mechanisms like
\DUrole{raw-tex}{\cite{O'Raifeartaigh:1975pr}}  of SUSY breaking  don't explicitely work in the MSSM because any non-zero v.e.v.s for \DUrole{raw-tex}{$F-$} and
\DUrole{raw-tex}{$D-$}  terms of the SM fields spoil \DUrole{raw-tex}{$U(1)$} or \DUrole{raw-tex}{$SU(3)$} gauge invariance. Thus there is a hidden sector, the weakest part
of the MSSM, which is spontaneously breaking via \DUrole{raw-tex}{$F-$} mechanism. The special fields \texttt{messengers} like gravitino, gaugino or
gauge bosons can mediate SUSY breaking from hidden to the visible sector.

There are several requirements imposed on the Higgs sector that should be taken into account for the SUSY generalization of SM:
%
\begin{itemize}

\item Higgs fields have non-zero v.e.v.s. Therefore they cannot be superpartners of quarks  and leptons since
this would induce the spontaneous violation of baryon and lepton number through Yukawa interaction.

\item One needs at least two complex chiral multiplets to give masses to Up and Down quarks.

\end{itemize}

The latter is due to the form of the superpotential and chirality of the matter superfields. The Yukawa interaction in
Standard Model should be invariant under \DUrole{raw-tex}{$U(1)$} gauge subgroup. This imposes the fixed form of the interaction terms
\DUrole{raw-tex}{$$\mathcal{L}_Y = (y_e)_{\alpha\beta}\bar{L}_e^{\alpha} H E_e^{\beta} + (y_d)_{\alpha\beta}\bar{Q}^{\alpha} H D^{\beta} $$}
with zero hypercharge \DUrole{raw-tex}{$Y=-Y_L+Y_{H}+Y_E=-(-1)+1-2=0$}. The up quarks generates masses via interaction with charge-conjugated
Higgs doublet \DUrole{raw-tex}{$H^C=i\sigma^2H$} with \DUrole{raw-tex}{$Y_{H^C}=-1$}
\DUrole{raw-tex}{$$  \mathcal{L}_Y^u= (y_u)_{\alpha\beta}\bar{Q}^{\alpha}H^C U^{\beta},\,\,\, Y=-1/3-1+4/3=0.$$}
However, in SUSY, \DUrole{raw-tex}{$H$} is a left chiral superfield. Hence the superpotential, which is constructed out of the  left chiral
superfields, can contain only \DUrole{raw-tex}{$H$}  but not \DUrole{raw-tex}{$H^C$} which is the right chiral multiplet.

Another reason for introducing the  second left-handed Higgs doublet \DUrole{raw-tex}{$H_2$}  is the cancelation of chiral anomalies. The
chiral anomalies  \DUrole{raw-tex}{$Tr(Y^3)\neq 0$} spoils gauge invariance   and, hence,  the renormalizability of the theory.
SM is free of such chiral anomalies \DUrole{raw-tex}{$$ Tr(\sum_{i=e_L,e_R,\nu_L,u_L,etc} Y_i^3)=0 $$} but if a chiral Higgs doublet  is introduced,
it contains \texttt{higgsinos}, which are  fermions which hypercharges are non zero: \DUrole{raw-tex}{$Tr(Y^3) \neq 0$} . To compensate them,
one has to add the second Higgs doublet with opposite hypercharge.


\subsection{3.1~~~Field Content%
  \label{field-content}%
}

Minimal Supersymmetric Model associates Standard Model bosons with new fermions and SM fermions with new bosons.
The another Higgs  doublet is added to cancel the chiral anomalies and to make massive  up quarks. The particle
content of the MSSM  appears as \DUrole{raw-tex}{\cite{MSSM2}} illustrated in Table \DUrole{raw-tex}{$\ref{tbl2}$} .

\setlength{\DUtablewidth}{\linewidth}
\begin{longtable}[c]{|p{0.114\DUtablewidth}|p{0.339\DUtablewidth}|p{0.197\DUtablewidth}|p{0.300\DUtablewidth}|}
\caption{Particle content of the MSSM \DUrole{raw-tex}{\label{tbl2}}}\\
\hline
\textbf{%
Superfield
} & \textbf{%
Bosons (spin 0,1)
} & \textbf{%
Fermions (spin 1/2)
} & \textbf{%
\DUrole{raw-tex}{$SU(3)_C$ \hfill $SU(2)_L$\hfill $U(1)_Y$}
} \\
\hline
\endfirsthead
\caption[]{Particle content of the MSSM \DUrole{raw-tex}{\label{tbl2}} (... continued)}\\
\hline
\textbf{%
Superfield
} & \textbf{%
Bosons (spin 0,1)
} & \textbf{%
Fermions (spin 1/2)
} & \textbf{%
\DUrole{raw-tex}{$SU(3)_C$ \hfill $SU(2)_L$\hfill $U(1)_Y$}
} \\
\hline
\endhead
\multicolumn{4}{c}{\hfill ... continued on next page} \\
\endfoot
\endlastfoot

\textbf{Gauge}
 & 

 & 

 & 

 \\
\hline

\DUrole{raw-tex}{$G_a$}
 & 
gluon, \DUrole{raw-tex}{$g_a$}
 & 
gluino, \DUrole{raw-tex}{$\tilde{g}_a$}
 & 
\DUrole{raw-tex}{8\hfill      0 \hfill       0}
 \\
\hline

\DUrole{raw-tex}{$V^k$}
 & 
weak,  \DUrole{raw-tex}{$W^k$}
 & 
wino, \DUrole{raw-tex}{$\tilde{W}^k$}
 & 
\DUrole{raw-tex}{1\hfill     3\hfill        0}
 \\
\hline

\DUrole{raw-tex}{$B$}
 & 
hypercharge, \DUrole{raw-tex}{$B$}
 & 
bino, \DUrole{raw-tex}{$\tilde{B}$}
 & 
\DUrole{raw-tex}{1\hfill     1\hfill        0}
 \\
\hline

\textbf{Matter}
 & 

 & 

 & 

 \\
\hline

\DUrole{raw-tex}{$L_i$}
 & 
sleptons, \DUrole{raw-tex}{$(\tilde{\nu},\tilde{l}_L)$}
 & 
leptons, \DUrole{raw-tex}{$(\nu,l_L)$}
 & 
\DUrole{raw-tex}{1\hfill     2\hfill        -1}
 \\
\hline

\DUrole{raw-tex}{$E_i^C$}
 & 
\DUrole{raw-tex}{\hfill $\tilde{l}_R^C$ \hfill}
 & 
\DUrole{raw-tex}{\hfill $l_R^C$ \hfill}
 & 
\DUrole{raw-tex}{1\hfill     1\hfill        2}
 \\
\hline

\DUrole{raw-tex}{$Q_i$}
 & 
squarks, \DUrole{raw-tex}{$(\tilde{u},\tilde{d})_L$}
 & 
quarks, \DUrole{raw-tex}{$(u,d)_L$}
 & 
\DUrole{raw-tex}{3\hfill     2\hfill        1/3}
 \\
\hline

\DUrole{raw-tex}{$U_i^C$}
 & 
\DUrole{raw-tex}{$\tilde{u}_R^C$}
 & 
\DUrole{raw-tex}{$u_R^C$}
 & 
\DUrole{raw-tex}{$\bar{3}$\hfill     1\hfill       -4/3}
 \\
\hline

\DUrole{raw-tex}{$D_i^C$}
 & 
\DUrole{raw-tex}{$\tilde{d}_R^C$}
 & 
\DUrole{raw-tex}{$d_R^C$}
 & 
\DUrole{raw-tex}{$\bar{3}$\hfill     1\hfill       2/3}
 \\
\hline

\textbf{Higgs}
 & 

 & 

 & 

 \\
\hline

\DUrole{raw-tex}{$H_1$}
 & 
Higgses, \DUrole{raw-tex}{$H_1$}
 & 
higgsinos, \DUrole{raw-tex}{$\tilde{H}_1$}
 & 
\DUrole{raw-tex}{1\hfill     $\bar{2}$\hfill        -1}
 \\
\hline

\DUrole{raw-tex}{$H_2$}
 & 
\DUrole{raw-tex}{$H_2$ \hfill {}}
 & 
\DUrole{raw-tex}{$\tilde{H}_2$\hfill {}}
 & 
\DUrole{raw-tex}{1\hfill     2\hfill        1}
 \\
\hline
\end{longtable}

The gauge (vector) superfields consist of the SM gauge bosons \DUrole{raw-tex}{$W^k,\,\, k=1,2,3$} and the gluon field \DUrole{raw-tex}{$g_a, \,\, a=1..8$}
accompanied  by their superpartners, spin 1/2 Majorana particles called \textbf{gauginos}  and gluino correspondingly.
Like the guage fields. these fermion fields transform as the adjoint representation of the appropriate group factor:
\DUrole{raw-tex}{$SU(3)_C,\,\, SU(2)_L\,\,\, U(1)_Y$} . Hereafter, tilde denotes a superpartner \DUrole{raw-tex}{$\tilde{l}$} of an ordinary particle, and
\DUrole{raw-tex}{$^C$} indicates on the charge-conjugation \DUrole{raw-tex}{$\psi_L^C = i\sigma_2\psi_R^{\star},\,\,\, \psi_R^C=-i\sigma_2\psi_L^{\star}$}

The chiral (matter) superfields content of the SUSY is exactly the same as in SM: three families of the chriral quarks and leptons.
Each family has five different gauge representations  of SM Weyl fermions \DUrole{raw-tex}{$\psi_L=(1/2,0),\,\, \psi_R=(0,1/2)$} shown as the
third column of the Table \DUrole{raw-tex}{\ref{tbl2}} :
\DUrole{raw-tex}{$$ Q_i =q_L(3,2)_{+1/3},\,\,U_i^C=u_R^C(\bar{3},1)_{-4/3},\,\, D_i^C=d_R^C(\bar{3},1)_{+2/3}, $$
$$L_i=l_L(1,2)_{-1},\,\, E_i^C=l_R^C(1,1)_{+2} $$}
In addition, the matter families in SUSY are populated  by five gauge representation of spin-0 particles,
scalars and pseudoscalars, with
the same quantum numbers of the electric charge \DUrole{raw-tex}{$Q$} and third projection of the weak isospin \DUrole{raw-tex}{$T_3$} as their SM partners have
\DUrole{raw-tex}{$$scalar(pseudo)\sim(0,1/2)\times (1/2,0)\sim \overline{\psi}\psi(\overline{\psi}\gamma^5\psi),$$} where
\DUrole{raw-tex}{$$ scalar=\psi_R^{\dag}\psi_L + \psi_L^{\dag}\psi_R,\,\,\,\,\, pseudo=\psi_R^{\dag}\psi_L - \psi_L^{\dag}\psi_R $$}
The scalar partners \DUrole{raw-tex}{$\tilde{l}_L$} and  \DUrole{raw-tex}{$\tilde{l}_R$} couples only to \DUrole{raw-tex}{$\tilde{l}_L\,\,(l_L)$} and
\DUrole{raw-tex}{$\tilde{l}_R\,\,(l_R)$} via gauge boson (gauginos) because the gauge interaction  conserves the chiral flavor.
However, there can be \DUrole{raw-tex}{$\tilde{l}_L-\tilde{l}_{R}$} mixing in MSSM, if one considers the  Yukawa interaction
in the superpotential which violates \DUrole{raw-tex}{$R-$} parity. Such terms in SM would violate fermion and
barion numbers. This mixing is generally negligible except for third generation of sfermions.

The presence of an extra Higgs doublet in SUSY model is a novel feature of the SM-like theory. In the MSSM, one has two doublets
\DUrole{raw-tex}{$H_1$} and \DUrole{raw-tex}{$H_2$} of the representions \DUrole{raw-tex}{$(1,2,-1)$} and \DUrole{raw-tex}{$(1,\bar{2},+1)$} accordingly
\DUrole{raw-tex}{$$ H_1=\left(\begin{array}{l}  H_1^0 & \\
H_1^- \end{array} \right)=\left( \begin{array}{l} v_1 + \frac{S_1^0(x) + iP_1^0(x)}{\sqrt{2}}  & \\ \frac{S_1^-(x) + iP_1^-(x)}{\sqrt{2}} & \end{array} \right),\,\,
H_2=\left(\begin{array}{l}
H_2^+ & \\
H_2^0 & \end{array} \right)=\left( \begin{array}{l}  \frac{S_2^+(x) + iP_2^+(x)}{\sqrt{2}} & \\ v_2 + \frac{S_2^0(x) + iP_2^0(x)}{\sqrt{2}}  &  \end{array} \right),\,\,
$$}
If the vacum state is not  \DUrole{raw-tex}{$SU(2)_L$} invariant any more, the three degrees of freedom \DUrole{raw-tex}{$P_1^0,P_1^-,P_2^+$}
are the Goldston modes \DUrole{raw-tex}{\cite{Goldstone:1961eq}, \cite{Goldstone:1962es}} which can
be gauged away \DUrole{raw-tex}{\cite{LopezOsorio:2004pu}} using unitary gauge  \DUrole{raw-tex}{\cite{Weinberg:1973ew}} . Hence there are only \DUrole{raw-tex}{$5=8-3$}
dynamical degrees of  freedom resulting into the five  massive physical states
\DUrole{raw-tex}{$$ h(x)\sim S_1^0, \,\,\, H(x)\sim S_2^0,\,\,\, H^-(x)\sim S_1^-,\,\,\, H^+(x)\sim S_2^+,\,\,\, A(x)\sim P_2^0(x),  $$}
where \DUrole{raw-tex}{$h(x)$} and \DUrole{raw-tex}{$H(x)$}  are CP even neuthral Higgs bosons, \DUrole{raw-tex}{$A(x)$} is a CP odd neutral Higgs boson and \DUrole{raw-tex}{$H^{\pm}(x)$} are
two charged Higgs fields. I will consider the mass eigenstates of the Higgs sector below.


\subsection{3.2~~~Lagrangian of the MSSM%
  \label{lagrangian-of-the-mssm}%
}

If supersymmetry is exact, superpartners of ordinary particles have the same masses and have to be observed. The absence of
them at modern energies is believed to be explained by the fact their masses are very heavy, that means that the supersymmetry
should be broken. Hence, if the energy of accelerators would be  high enough, the superpartners could be created during
particle collisions.

The Lagrangian of the MSSM consists of two parts. The first part is SUSY generalization of the Standard Model, while
the second one represents the SUSY breaking.  Recalling \DUrole{raw-tex}{$(\ref{eq37})$} and extending the gauge group to
\DUrole{raw-tex}{$SU(3)_C\times SU(2)_L\times U(1)_Y$} \DUrole{raw-tex}{$$ \Phi^{\dag}e^{gV}\Phi \Rightarrow \Phi^{\dag}e^{g_3V_3+g_2V_2+g_1V_1}\Phi $$}
someone can  write

\begin{flalign}\label{eq51}
\mathcal{L} = \mathcal{L}_{SUSY}  + \mathcal{L}_{breaking},\nonumber \\
\mathcal{L}_{SUSY} = \mathcal{L}_{gauge} + \mathcal{L}_{Yukawa},\nonumber \\
\mathcal{L}_{gauge} = \sum_{SU(3),SU(2),U(1)}\frac{1}{4}(\int d^{2}\theta TrW^{\alpha}W_{\alpha}+\int d^{2}\overline{\theta}Tr\overline{W}^{\alpha}\overline W_{\alpha}) + \nonumber \\
\sum_{matter}\int d^{2}\theta d^{2}\overline{\theta}\Phi_{i}^{\dagger}e^{g_{3}\hat{V}_{3}+g_{2}\hat{V}_{2}+g_{1}\hat{V}_{1}}\Phi_{i},\nonumber \\
\mathcal{L}_{Yukawa} = \int d^{2}\theta \mathcal{W}_{R}+h.c
\end{flalign}

\DUrole{raw-tex}{\noindent}
In terms of the component fields, the Lagrangians \DUrole{raw-tex}{$\mathcal{L}_{gauge}$}  and \DUrole{raw-tex}{$\mathcal{L}_{Yukawa}$} introduced  above  \DUrole{raw-tex}{$(\ref{eq51})$} take a familiar form

\begin{flalign}\label{eq52}
\mathcal{L}_{gauge} = \sum_{a=SU(3),SU(2),U(1)}(-\frac{1}{4}F_{\mu\nu}^{a}F^{a\mu\nu}-i\lambda^{a}\sigma^{\mu}D_{\mu}\overline{\lambda}^{a}-\frac{1}{2}D^{a}D^{a} +   \nonumber \\
 D_{\mu}\phi_i^{\star}D^{\mu}\phi_i + i \xi^{\dag}\sigma^{\mu}D_{\mu} \xi - i\sqrt{2}g^a ( \phi_i^{\star}  T^a\phi_i \lambda_a^T\xi_i - \xi^{\dagger}_iT^a\lambda_a^{\star}\phi_i)   \nonumber \\
 \mathcal{L}_{Yukawa}=-F_i^{\dagger}F_i  - (\frac{1}{2} \xi_{i}^T \frac{ \partial^2  \mathcal{W}_R } {\partial \phi_i\partial \phi_j}\xi_j + h.c.)
\end{flalign}

\DUrole{raw-tex}{\noindent}
The equations of motions for  the auxilary fields \DUrole{raw-tex}{$F_i$} and \DUrole{raw-tex}{$D_a$}
\DUrole{raw-tex}{$$ F_i^{\star}=-\frac{\partial \mathcal{W}_R} {\partial \phi_i},\,\,\, D_a=-g^a\phi_i(T^a)_{ij}\phi_j $$}
were used to derive \DUrole{raw-tex}{$(\ref{eq52})$} .
I will focus on the SUSY generalization of Yukawa interaction \DUrole{raw-tex}{$\mathcal{W}_R$} which preserves the \DUrole{raw-tex}{$R-$} parity:

\begin{flalign}\label{eq53}
\mathcal{W}_{R}=\epsilon_{ij}(y_{ab}^{U}Q_{a}^{j}U_{b}^{C}H_{2}^{i}+y_{ab}^{D}Q_{a}^{j}D_{b}^{C}H_{1}^{i}+y_{ab}^{L}L_{a}^{j}E_{b}^{C}H_{1}^{i}+\mu H_{1}^{i}H_{2}^{j}),
\end{flalign}

\DUrole{raw-tex}{\noindent}
where \DUrole{raw-tex}{$y^{U,D,L}_{ab}$} are the \DUrole{raw-tex}{$3\times 3$} matrices  and \DUrole{raw-tex}{$Q_i$, $L_i$, $U^C_i$, $D_i^C$} are the supermultiplets given by the Table \DUrole{raw-tex}{${\ref{tbl2}}$} .
It is easy to show using \DUrole{raw-tex}{$(\ref{eq33})$, $(\ref{eq34})$ , $(\ref{eq366})$} that  \DUrole{raw-tex}{$R-$} parity invariance of \DUrole{raw-tex}{$\mathcal{W}_R$}
requires to have for any terms in the superpotential  \DUrole{raw-tex}{$$ R_p(W_R)=(-1)^{2k}, $$}  where \DUrole{raw-tex}{$k$} is some an integer
number. This leads to the following phenomenological predictions:
%
\begin{itemize}

\item Sparticles  can only be paired produced

\item Sparticles decay to SM particles and an odd number of sparticles

\item A chain of sparticles decays must be finalized by the production of the  stable Lightest Supersymmetric Particle (LSP)

\end{itemize}


\subsection{3.3~~~Breaking SUSY in the MSSM%
  \label{breaking-susy-in-the-mssm}%
}

As I mentioned previously, SUSY of MSSM can't be broken by developing non-zero v.e.v.s of some fields, i.e. the auxilary \DUrole{raw-tex}{$F-$}  fields,
without spoiling the gauge invariance. The most common scenario for producing low-energy supersymmetry breaking is   the
mediation of the  breaking from the hidden sector by means of the massive messengers \DUrole{raw-tex}{\cite{SUSYbreaking}} \DUrole{raw-tex}{$X$} with \DUrole{raw-tex}{$$M_X\sim <0|F_X|0> \sim M_{SUSY}\times M_{Pl}\sim 10^{22} GeV$$}
The effective theory of SUSY with the hidden sector contains higher dimension operators \DUrole{raw-tex}{$W \sim  X^{\dag}X\Phi_i^{\dag}\Phi_i$}
\DUrole{raw-tex}{$$ \Phi_i^{\dag}\Phi_i + X^{\dag}X+\frac{c_i}{M_{Pl}}X^{\dag}X\Phi_i^{\dag}\Phi_i,$$}
which give mass terms of the chiral superfield \DUrole{raw-tex}{$\Phi_i$} when all heavy states \DUrole{raw-tex}{$X$} are integrated out:

\begin{flalign}\label{eq53}
M_X^2\sim <0|F_X|0>,\,\,\,\frac{c_i}{M_{Pl}}X^{\dag}X\Phi_i^{\dag}\Phi_i  \Rightarrow M_{\Phi_i}^2 \Phi_i^{\dag}\Phi_i,\,\,\, M_{\Phi_i}^2 = c_i \frac{M_X}{M_{Pl}}
\end{flalign}

\DUrole{raw-tex}{\noindent}
The SUSY breaking \DUrole{raw-tex}{$$ <0|F_X|0>\neq 0 $$} in the hidden sector is propogated  to the visible sector, MSSM, and breaks it.
The   messenger  \DUrole{raw-tex}{$X$} may be a spin 3/2 particle, \emph{gravitino}  in the gravity mediation (\textbf{SUGRA})
breaking scenario \DUrole{raw-tex}{\cite{SUGRA}}. Or \DUrole{raw-tex}{$X$} might be a SM-singlet with non-zero v.e.v.s which couples to the chiral superfields \DUrole{raw-tex}{$Q_3$} and  \DUrole{raw-tex}{$\bar{Q}_3$} to be
\DUrole{raw-tex}{3 and $\bar{3}$} representations of  SU(3) \DUrole{raw-tex}{$$ W\sim X\bar{Q}_3Q_3.$$}    \DUrole{raw-tex}{$Q_3 (\bar{Q}_3)$} chiral multiplets become
the messengers dynamically breaking the supersymmetry at the low energy scales \DUrole{raw-tex}{$<F_X> << M_{Pl}$}. Masses of the gauginos  and
sfermions in the scenario of gauge mediated SUSY breaking (\textbf{GMSB} ) are the one-loop and two-loop effects initiated by
\DUrole{raw-tex}{$Q_3(\bar{Q}_3)$} messengers.

These mechanisms  can be  phenomenologicaly incorporated  into the MSSM  as some extensions of the Lagrangian
via the effective high dimension operator \DUrole{raw-tex}{$\mathcal{L}_{soft}$} after integration of  all messengers \DUrole{raw-tex}{$X$} out.

.

\begin{flalign}\label{eq54}
\mathcal{L}_{soft}= -\sum_{i}M_{i}^{2}|\phi_{i}|^{2}-(\sum_{\alpha}M_{\alpha}\tilde{\lambda}^C_{\alpha}\tilde{\lambda}_{\alpha}-B\epsilon_{ab}H^a_{1}H^b_{2} \nonumber \\
 A_{ab}^{U}\tilde{Q}_{a}\tilde{U}_{b}^{c}H_{2}+A_{ab}^{D}\tilde{Q}_{a}\tilde{D}_{b}^{c}H_{1}+A_{ab}^{L}\tilde{L}_{a}\tilde{E}_{b}^{c}H_{1}+h.c.)
\end{flalign}

\DUrole{raw-tex}{\noindent}
The bilinear and trilinear couplings \DUrole{raw-tex}{$B$ and $A_{ab}$} are such that gauge invariance is not broken. The Lagrangian \DUrole{raw-tex}{$(\ref{eq54})$}
is only possible choice that does not destroy the renormalizability of the theory \DUrole{raw-tex}{\cite{renorm}}.
The terms \DUrole{raw-tex}{$(\ref{eq54})$} provide the mass splitting between particles living in
the same supermultiplet and ensure the high  masses of
sparticles. \DUrole{raw-tex}{$\mathcal{L}_{soft}$}  introduces 104 additional parameters which are purely phenomenological.
I'll focus on the Higgs
sector of MSSM and show how  Electroweak Symmetry breaking (\textbf{EW})   is dynamically achieved
with help of Renormalization Group
(\textbf{RG}) equations of the Higgs sector.


\subsection{3.4~~~The Higgs sector and Electroweak Symmetry Breaking in the MSSM%
  \label{the-higgs-sector-and-electroweak-symmetry-breaking-in-the-mssm}%
}

The higgs sector

